PARACOMPACTNESS AND SEPARATELY ...

PARACOMPACTNESS AND SEPARATELY CONTINUOUS MAPPINGS V. K. MASLYUCHENKO† , O. V. MASLYUCHENKO‡ , V. V. MYKHAYLYUK§ , AND O. V. SOBCHUK¶ Abstract. In this paper the results of the Separately Continuous Mappings Theory which obtained by using of the paracompactness, locally finite systems and partitions of unit are considered. Key words. Paracompact space, separately continuous mapping, Baire classification, inverse problem. AMS subject classifications. 54C05, 54C30

1. Introduction. In 1944 J. Dieudonné [8] introduced a notion of the paracompactness and established that metrizable separable or locally compact spaces are paracompact. As well known the main means in the definition of the paracompactness are locally finite coverings which were gave in 1924 by P. Alexandroff [1]. In fact, he noted that a separable metrizable space is paracompact. The paracompactness of arbitrary metrizable spaces were established by A. Stone [43]. K. Morita [35] proved the paracompactness of Lindelöff spaces. After that E. Michael [33] introduced the partition of unit which became invariable tools of Topology and Analysis. Investigations of separately continuous mappings were begun in classical R. Baire [2], H. Lebesgue [15,16] and H. Hahn [11,12,13] papers and they were continued in the numerous papers of mathematicians of XX century (see [39,40,30]). These investigations are divided into two directions: a study of the size of joint continuity points set of separately continuous mappings (the direct and the inverse problems) and Baire and Lebesgue classifications of separately continuous mappings. W. Rudin [41] was the first who used the Stone theorem and partitions of unit in the investigation of separately continuous mappings, Baire classification and with their help he essentially improved the classical H. Lebesgue and H. Hahn results. However, J. Saint-Raymond [42] had already used partitions of unit with the same purpose a little bit earlier but only in the case of metrizable compact which does not require using of the Stone theorem. R. Hansell [14] used the presence of σ-locally finite base in metrizable spaces and the Stone theorem. Beginning with paper [25] locally finite systems, the Stone theorem and partitions of unit were intensively used by the authors as in the solving of the inverse problems of the Separately Continuous Mappings Theory as in the investigation of Baire and Lebesgue classifications. This paper sums up the results of these investigations.

† Department of Mathematical 58000 Chernivtsi, Ukraine; e-mail: ‡ Department of Mathematical 58000 Chernivtsi, Ukraine; e-mail: § Department of Mathematical 58000 Chernivtsi, Ukraine; e-mail: ¶ Department of Mathematical 58000 Chernivtsi, Ukraine; e-mail:

Analysis, Chernivtsi state university, [email protected] Analysis, Chernivtsi state university, [email protected] Analysis, Chernivtsi state university, [email protected] Analysis, Chernivtsi state university, [email protected] 147

vul. Kotsyubyns’koho, 2, vul. Kotsyubyns’koho, 2, vul. Kotsyubyns’koho, 2, vul. Kotsyubyns’koho, 2,

148 V.K.MASLYUCHENKO, O.V.MASLYUCHENKO, V.V.MYKHAYLYUK, O.V.SOBCHUK 2. Inverse problems. 2.1. Approximating theorems. The notion of locally finite system or family of sets will play the main role in our constructions. Here we shortly recall conceptions and results concerning it. A system A of sets in a topological space X is called locally finite at a point x ∈ X if there exists a neighborhood U of x in the space X which intersects only with finite number of sets from A, i.e. for this neighborhood U the system A(U ) = {A ∈ A : T A U 6= Ø} is finite. We say that a system A is locally finite on a set E ⊆ X, if it is locally finite at every point x ∈ E. If a system A is locally finite on the whole space X then it is called locally finite. Analogous notions possible to introduce for families of sets. Recall that by a family of sets in X we mean a mapping α : S → 2X , which assigns to every element s from S a set As = α(s) which is a subset of X. Such a family α is written in the form α = (As : s ∈ S). To every family of sets α = (As : s ∈ S) it is compared a system of sets A = α(S) = {As : s ∈ S}. A family α = (As : s ∈ S) of subsets of a topological space X is called locally finite at a point T x if there exists its neighbourhood U in X such that the set S(U ) = {s ∈ S : As U 6= Ø} is finite. If a family α = (As : s ∈ S) is locally finite at a point x then so is the corresponding system A = {As : s ∈ S} but converse is not correct. The notion of finitness or locally finitness on a set for families is introduced in the same way as for a system. A system A and a family α = (As : s ∈ S) are called pointwise finite in X if for every point x ∈ X the sets A(x) = {A ∈ A : x ∈ A} and S(x) = {s ∈ S : x ∈ As } are finite respectively. A system A of sets is called inscribed in a system B of sets if for every A ∈ A there exists B ∈ B such that A ⊆ B. A family α = (As : s ∈ S) of sets is inscribed in a family β = (Bt : t ∈ T ) of sets if the corresponding system A = {As : s ∈ S} is inscribed in the corresponding system B = {Bt : t ∈ T }. We shall use also expressions ”the family α is inscribed in the system B” or ”the system A is inscribed in the family β” which have the obvious sense. It is said that a family α = (As : s ∈ S) combinatorial inscribed in a family β = (Bs : s ∈ S), if As ⊆ Bs for every s ∈ S. A topological space X is called paracompact if in every its open covering it possible to inscribe some open locally finite covering. We shall use the Stone theorem [10, p.414; p.444] on paracompactness of metrizable spaces. Note that every Hausdorff paracompact space (or shortly paracompactum) is a normal space [10,p.445]. A family (ϕs : s ∈ S) of functions ϕs : X → R is called locally finite (at a point, on a set) if so is the family (suppϕs : s ∈ S) of supports suppϕs = {x ∈ X : ϕs (x) 6= 0} of these functions. It is said that a family (ϕs : s ∈ S) is subordinated to a system of sets A or to a family of sets α, if the family of supports (suppϕs : s ∈ S) is inscribed in A or in α. A family (ϕs : s ∈ S) P of continuous functions ϕs : X → [0, 1] is called partition of unit on the space X if ϕs (x) = 1 for every x ∈ X. Notice that for every open s∈S

covering of paracompactum X there exists a locally finite partition of unit which is subordinated to it [10,p.447]. Approximate theorems are the main technical tool in the solving of the inverse problems. Let S be a subset of a topological space X. A sequence of families τn : S → 2X is called localized at a point x ∈ X, if for every neighborhood U of x there T exist a neighborhood U0 of x and a number n0 such that τn (s) Un = Ø whenever s ∈ S \ U and n > n0 . If a sequence of families τn is localized at every point of X then we say that it is localized.

PARACOMPACTNESS AND SEPARATELY CONTINUOUS MAPPINGS

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Let X be a metric space. Put δ(A) = sup{diamA : A ∈ A} for a nonempty system A of nonempty sets in X. By the Stone theorem it easy to construct a sequence of locally finite open coverings Gn of X such that δ(Gn ) → 0 as n → ∞. Just that very result is used in the proofs of approximate theorems. Recall that a mapping π : S → X is called a choice function for a family τ : S → 2X of nonempty subsets in X, if π(s) ∈ τ (s) for every s ∈ S. Theorem 2.1. For every nowhere dense closed set S in a metrizable space X with the topology T there exists a localized sequence of families τn : S → T (X \ S) of open sets in X which satisfies the following conditions: (i) for every n the system Un = τn (S) is locally finite; ∞ S (ii) the system U = Un is pointwise finite and locally finite on the complement n=1

X \ S; (iii) equalities lim πn (s) = s, s ∈ S hold for every sequence of choice functions n→∞ πn : S → X for τn then. Proof. Fix some metric on the space X. Given a sequence of locally finite open coverings Gn of the space X for which δ(Gn ) → 0 as n → ∞. For every s ∈ S the system Gn (s) = {G ∈ Gn : s ∈ G} is not empty since Gn covers X. Let ϕn be a choice function for the family (Gn (s) : s ∈ S). Thus s ∈ Gn,s = ϕn (s) ∈ Gn for every pair (n, s) ∈ N × S. Put τn (s) = Un,s = Gn,s \ S. Since Gn,s is open nonempty and S is closed nowhere dense then all sets Un,s are nonempty and open. Then Un,s ⊆ X \ S for any n ∈ N and s ∈ S. Thus we have defined some family τn : S → T (X \ S) for every n. The sequence of these families τn is to be found. Localized sequence of families τn : S → T (X \ S) which satisfies the conditions (i)–(iii) we shall call approximating for the set S. A system B is called dense in X, if every open nonempty subset of X contains some nonempty set from B. A dense system of open sets is called pseudobase. Let B be some system of open subsets of X. A set S ⊆ X is called B-approximating, if there exist a family π : S →S2X of subsets of X and a disjoint family τ : P → B which is defined on the body P = π(S) of the family π such that p ∈ τ (p), τ (p) ∩ S 6= Ø, s ∈ π(s) for p ∈ P and s ∈ S andSthe system τ (PE ) is locally finite outside E for an arbitrary E ⊆ S where PE = π(E). If in addition there exists a mapping ν : P → N such that for any s ∈ S, any neighborhood U of s and n ∈ N there exists p ∈ π(s) ∩ U such that ν(p) > n, then the set S is called strongly B-approximating. It is easy to see that these notions coincide if S is dense in some Gδ -set. And also if S is B-approximating and a set P from the definition of B-approximability is countable, then S is strongly B-approximating. Specifically, both these notions are equivalent for perfect spaces. Theorem 2.2. Let X be a metrizable space, B be its pseudobase and S be nowhere dense in X. Then S is B-approximating. The construction of corresponding families π : S → 2X and τ : P → B is performed like as in the proof of Theorem 1. Let X = X1 × · · · × Xn be the product of topological spaces and E ⊆ X. For a point x = (x1 , ..., xn ) ∈QX and an arbitrary index i = 1, ..., n put x î = î = ˆ i by qi (x) = (x1 , ..., xi−1 , xi+1 , ..., xn ), X Xk and define projection qi : X → X k6=i

x î . We say that a set E is projectively nowhere dense (meager) with respect to i-th ˆ i . If for every point x ∈ X there exists axis, if qi (E) is nowhere dense (meager) in X its neighborhood U in X such that E ∩ U is projectively nowhere dense (meager) with respect to i-th axis, then the set E is called locally projectively nowhere dense (mea-

150 V.K.MASLYUCHENKO, O.V.MASLYUCHENKO, V.V.MYKHAYLYUK, O.V.SOBCHUK ger) with respect to i-th axis. A projectively nowhere dense (meager) with respect to every axis set is called projectively nowhere dense (meager). The same concerns to local varieties of these notions. Theorem 2.3. Let the product X = X1 × · · · × Xn be a paracompact space and E be a locally projectively meager set in X. Then E can be represented in the form of the union of a sequence of locally projectively nowhere dense sets En . If in addition E is Fσ -set, then the sets En could be found to be closed. n S qi−1 (qi (E) is called a cross of a subset E in the product X. We A set xpE = i=1

say that a set M ⊆ X is a cross-neighborhood of a point x, if U ∩ xp{x} ⊆ M for some neighborhood U of x. A set M ⊆ X is called cross-neighborhood of a set E, if M is cross-neighborhood of all its points. We say that a system A is crosswise locally finite at a point x if there exists cross-neighborhood of x such that it intersects with finite numbers of elements of A only. A system A is called crosswise locally finite, if it is so at every point of X. Theorem 2.4. Let X = X1 ×· · ·×Xn be the product of topological spaces, E ⊆ X and one of the following conditions is satisfied: (i) X is a paracompact space and E is a locally projectively nowhere dense set; (ii) X is a metrizable space and E is a nowhere dense set, the closure of which is the union of a sequence of locally projectively nowhere dense sets; (iii) X is a hereditary paracompactum and every nonempty closed subset E is projectively nowhere dense at least at one of its point. Then there exists a nowhere dense cross-neighborhood of E. A set E of the product X is called (strongly) oblique approximating, if there exists crosswise locally finite system B of open sets such that S is (strongly) Bapproximating. Theorem 2.5. Let a subset E of the product X be B-approximating for any pseudobase B in X and E has nowhere dense cross-neighborhood M . Then E is oblique approximating. 2.2. General inverse problem. The general inverse problem of Separately Continuous Mappings Theory is that for a given set E which lies in the product X of topological spaces X1 , ..., Xn to construct a separately continuous mappings f : X → R such that the discontinuity points set D(f ) coincides with E. The direct theorems on a smallness of D(f ) for separately continuous functions give necessary conditions on the set E such that the inverse problem has a solution. The solution of the inverse problem under given necessary conditions results the complete description of the discontinuity points set of separately continuous functions. There are a lot of results giving complete description. So, R.Keshner in [19] given a characterization of the set D(f ) for separately continuous functions which in the case n = 2 looks so: a set E ⊆ R2 is the discontinuity points set of some separately continuous function f if and only if it is the projectively meager Fσ -set. J.Breckenridge and T.Nishiura in [5], using absolutely other approach, have solved the inverse problem for projectively meager Fσ -sets, which lie in products of two metrizable spaces. By the Calbrix-Troallic theorem for second countable spaces X and Y and a separately continuous functions f : X × Y → R the set D(f ) is projectively meager. Since D(f ) is an Fσ -set for realvalued functions then we receive a description of discontinuity points set of separately continuous functions on products of two separable metrizable spaces which is similar to the Keshner description. In particular, it is on products of two metrizable compact.


151

New methods of solving of the inverse problem on products of metrizable spaces were offered in [26,29,25]. They give similar description in the separable case. The method of [25] there was especially perspective. It used locally finite systems and the Stone theorem. It was advanced in [30,36] and was applied there to a wide class of spaces, which are called favorable. Besides, by the theorem on dependence of separately continuous functions on countable number of coordinates, in [27] it was possible to receive a description of sets D(f ) for separately continuous functions f : X × Y → R, where X and Y are products of families of metrizable compacts. The Keshner characterization is not true for separately continuous functions on products of any metrizable spaces. An example of separately continuous functions with the projection to the first multiplicand which equals to R was constructed in [23] (see bellow). Though the projection of the set D(f ) is globally large but it is locally small in this example. Namely the projection to the first multiplicand of any set D(f ) ∩ (R × V ) where V is a ball in l∞ of rather small radius consisting of no more than one point. It shows that the smallness of the set D(f ) for separately continuous functions on products of metrizable spaces may be described in the local terms. The sets D(f ) were described by the technique of favorable spaces where spaces X and Y are metrizable and a function f : X × Y → R is separately continuous (see [30]). The general inverse problem for products of n spaces was solved in [28] and we shall state it in this item. Moreover, a description of discontinuity points sets of separately continuous functions will be received in this case. At first we shall construct a separately continuous function f : R × l∞ → R with prR (D(f )) = R. We shall receive a more general statement using the following result. Theorem 2.6. Let X and Y be completely regular topological spaces, a be a nonisolated Gδ -point in X, b be a nonisolated Gδ -point in Y and Y be locally connected or b has a countable neighborhood base in Y . Then there exists a separately continuous function f : X × Y → R such that D(f ) = {(a, b)}. Proof. The desired function is defined by the formula f (x, y) = g(ϕ(x), ψ(y)) where ϕ : X → [0, 1], ψ : Y → [0, 1] are continuous functions such that ϕ−1 (0) = {a}, ψ −1 (0) = {b} and g : R2 → R is chosen by the suitable way to be separately continuous function with D(g) = {(0, 0)}. Theorem 2.7. Let X be a completely regular space every point of which is Gδ and nonisolated and Y be a completely regular space which has dense set of nonisolated points with countable neighborhood base. Suppose that there exists a locally finite family (Vx : x ∈ X) of nonempty open sets Vx . Then there exists a separately continuous functions f : X × Y → R with prX (D(f )) = X. Proof. Choose a nonisolated point yx ∈ Vx with countable neighborhood base. Then by Theorem 2.6 for every point x ∈ X there exists a separately continuous function hx : X × Y → R with D(hx ) = {(x, yx )}. Since Y is completely regular then for every x ∈ X there exists a continuous function gxP: Y → [0, 1] such that gx (yx ) = 1 and gx (y) = 0 on Y \ Vx . The function fu (x, y) = gu (y)hu (x, y) is to be found. u∈X

Notice that the conditions of Theorem 2.7 are satisfied if X is a Tikhonoff space of cardinality ℵ without isolated points and every point of X is Gδ -point, and Y is normed space which has a family (yi : i ∈ I) such that kyi − yj k ≥ ε > 0 for some ε and for every i 6= j, besides |I| = ℵ. Indeed, open balls B(yi , ε/4), i ∈ I are formed a locally finite family. A reindexation of this family gives needed family (Vx : x ∈ X). For example, if X = R and Y = l∞ then we can consider the family (yτ : τ ∈ 2N ) of characteristic sequences yτ of all subsets τ ⊆ N for which kyτ − yσ k = 1 whenever τ 6= σ and |2N | = 2ℵ0 = |R|.

152 V.K.MASLYUCHENKO, O.V.MASLYUCHENKO, V.V.MYKHAYLYUK, O.V.SOBCHUK Following simple lemmas are frequently used on the final stage of a solving of the inverse problem. Lemma 2.8. If f, g : X → R are lower semicontinuous functions then D(f +g) = S D(f ) D(g). Lemma 2.9. If un : X → R are lower semicontinuous functions and the series ∞ ∞ P S un (x) is uniformly convegent on X then D(f ) = D(un ). f (x) = n=1

n=1

Lemma 2.10. Let X1 , ..., Xn be topological spaces, X = X1 × · · · × Xn be the topological product and (fi : i ∈ I) be locally finite family of lower semicontinuous P separately continuous functions fi : X → R. Then the formula f (x) = fi (x) i∈I

defines aSlower semicontinuous separately continuous function f : X → R such that D(f ) = D(fi ). i∈I

Let us proceed to a solving of the general inverse problem on products of metrizable spaces. Lemma 2.11. Let X = X1 ×· · ·×Xn be the product of metrizable spaces and E be a closed projectively nowhere dense set in X. Then there exists a lower semicontinuous separately continuous function g : X → [0, +∞) such that D(g) = E. n S Proof. Put Ei = qi (E), Fi = Ei , Si = qi−1 (Fi ) and S = Si . Since E is i=1

projectively nowhere dense then every set Ei and its closure Fi are nowhere dense in ˆ i . Then sets Si are closed and nowhere dense in X. Thus the finite union S of these X sets is the same. It is easy to see that E ⊆ S. By the first approximating theorem there exists a sequence of families τm that approximates S. Put Vm = τm (E) and ∞ S V = Vm . Let assign to V ∈ V some point p(V ) ∈ V and a continuous function m=1 P ϕV (x) ϕV : X → [0, 1] such that ϕV (p(V )) = 1 and suppϕV ⊆ V . Put gm (x) = i g(x) =

∞ P m=1

V ∈Vm

gm (x). The function g is to be found.

Lemma 2.12. Let E be a closed locally projectively nowhere dense set in the product X = X1 × · · · × Xn of metrizable spaces. Then there exists a lower semicontinuous separately continuous function f : X → [0, +∞) such that D(f ) = E. Proof. It follows from the Stone theorem that there exists a locally finite open T covering U of X such that the set E U is projectively nowhere dense in X for T every U ∈ U. Then the closure E U is closed projectively nowhere dense in X for every U . By Lemma 2.11, there exists a lower semicontinuous separately continuous T function gU : X → [0, ∞) such that D(gU ) = E U . Since the space X is perfectly normal then for every U ∈ U there exists a continuous function hU : X → [0, 1] such that U = h−1 U ((0, 1]). The functions fU = gU hU are lower semicontinuous separately continuous and suppfU ⊆ U . Then P the family (fU : U ∈ U) is locally finite and by Lemma 2.10 the function f = fU is lower semicontinuous separately continuous U ∈U S too. It is nonnegative and D(f ) = D(fU ). Show that D(f ) = E. Clearly, U ∈U T D(fU ) ⊆ D(gU ) = E U ⊆ E = E for every U ∈ U. Hence D(f ) ⊆ E. Given x0 ∈ E. Since U is a covering of X then T there exists U0 ∈ U suchTthat x0 ∈ U0 . Then T hU0 (x) > 0 on U0 . Therefore D(gU0 ) U0 ⊆ D(fU0 ). But x0 ∈ E U0 ⊆ D(gU0 ) U0 and D(fU0 ) ⊆ D(f ). Hence x0 ∈ D(f ) and the proof is complete. Theorem 2.13. For every Fσ -set E in the product X = X1 × · · · × Xn of metrizable spaces, which is the union of a sequence of locally projectively meager Fσ -


153

sets there exists a separately continuous function f : X → R such that D(f ) = E. Proof. There exists a sequence of closed projectively nowhere dense sets Fm ∞ S in X such that E = Fm . By Lemma 2.12 for every Fm we can construct a m=1

lower semicontinuous separately continuous function fm : X → [0, +∞) such that D(fm ) = Fm . Chose some homeomorphism ϕ : [0, +∞) → [0, 1) and put um = 2−m ϕ ◦ fm . Functions um : X → R are lower semicontinuous separately continuous and D(um ) = Fm for they. Besides, 0 ≤ um (x) ≤ 2−m on X. Hence the series ∞ P um (x) is uniformly convergent on X. Then its sum f is lower semicontinuous

m=1

separately continuous on X. By Lemma 2.9 D(f ) =

∞ S m=1

D(um ) =

∞ S m=1

Fm = E.

Notice that locally projectively meager Fσ -sets need not be projectively meager. Indeed, set E0 of discontinuity points of f : R × l∞ → R (see above) is locally projectively meager, but prR (E0 ) = R. Thus, Theorem 2.13 is much stronger than the corresponding Breckenridge and Nishiura result [5]. Besides it is easy to construct an example of Fσ -set E, which is the union of a sequence of locally projectively meager Fσ -sets, and itself is not locally projectively meager. For this it is enough to put ∞ S m−1 E0 . E= n=1

If P is some property of mappings then by P (X, Y ) we denote the set of all mappings f : X → Y with the property P . By the letter C we denote the continuity property, and by the letter K we denote the quasi-continuity property (see [38]). Let X be some set and Y be a metric space. By |y 0 − y 00 |Y we shall denote the distance between points y 0 , y 00 ∈ Y . For a mapping f : X → Y and for a nonempty set A ⊆ X the number ωf (x) = sup{|f (x0 ) − f (x00 )|Y : x0 , x00 ∈ A} = diamf (A) is called oscillation of f on the set A. If X is a topological space and Ux is the system of all neighborhoods of x in X then the number ωf (x) = inf{ωf (U ) : U ∈ Ux } is called oscillation of f at the point x. The oscillation ωf : X → [0, +∞] is upper semicontinuous function. Hence the set Dε (f ) = {x ∈ X : ωf (x) ≥ ε} is closed in X for every ε. Clearly, C(f ) = {x ∈ X : ωf (x) = 0} and D(f ) = {x ∈ X : ωf (x) > 0}, where C(f ) is the continuity points set and D(f ) is the discontinuity points set of a mapping f . In particular, it implies that C(f ) is an Gδ -set and D(f ) is an Fσ -set for a mapping f : X → Y , where Y is a metrizable space. Put f x (y) = fy (x) = f (p) for arbitrary mapping f : X × Y → Z and a point p = (x, y) ∈ X × Y . Let us consider sets XQ (f ) = {x ∈ X : f x ∈ Q(Y, Z)}

and YP (f ) = {y ∈ Y : fy ∈ P (X, Z)}

for properties P and Q and a mapping f : X × Y → Z. By P Q(X × Y, Z) we denote the set of all mappings f : X × Y → Z, for which XQ (f ) = X and YP (f ) = Y . If X or Y are topological spaces then P Q(X × Y, Z) and P Q(X × Y, Z) mean the set of all mappings for which XQ (f ) = X and YP (f ) = Y or XQ (f ) = X and YP (f ) = Y respectively. We shall show that the sufficient conditions proved in Theorem 2.13 for solving of the general inverse problem, actually are necessary. The quasi-continuity is an important tool at the solving of the direct problem for mappings of many variables. According to the general denoting the set KC(X × Y, Z) is the set of all quasicontinuous with respect to the first variable and continuous with respect to the second variable mappings f : X × Y → Z.

154 V.K.MASLYUCHENKO, O.V.MASLYUCHENKO, V.V.MYKHAYLYUK, O.V.SOBCHUK Theorem 2.14. Let X be a Baire space, Y be a metric space, Z be a metrizable space and f ∈ KC(X × Y, Z). Then there exist a sequence of Fσ -sets En in X × Y ∞ S such that D(f ) = En and for any nonempty open in Y set V with diamV < 1/n n=1 T the projection prX (En (X × V )) is a meager set in X. Proof. Fix some metric on Z generating its topology. Denote by V (y0 , ε) the open ball in Y with the center at a point y0 and the radius equals to ε. Put Em,n = {(x, y) ∈ X × Y : ωf (x, y) ≥ ∞ S

for any m, n ∈ N. The sets En =

m=1 X1 , ..., Xn , Z

1 m

and ωf x (V (y,

1 1 )) < } n 4m

Em,n is to be found.

Theorem 2.15. Let be metrizable spaces and f : X1 ×· · ·×Xn → Z be a separately continuous mapping. Then the set E = D(f ) is the union of a sequence of locally projectively meager Fσ -sets. Proof. Fix some index i = 1, ..., n and prove that the set E is the union of a sequence of locally projectively meager with respect to i-th axis Fσ -sets Ei,m . By the ˆ i is the disjoint union of an open meager Banach theorem on category the space X set H, a closed nowhere dense set F and an open set G, which is Baire space with ˆ i topology. The open set H is an Fσ -set in the metrizable space X ˆ inductived from X T −1 S i. Since E is Fσ -set in X and the projection S qi is continuous then Ei,0 = E qi (H F ) is an Fσ -set in X. Besides qi (Ei,0 ) ⊆ H F . Hence Ei,0 is projectively meager with respect to the i-th axis. ˜ = q −1 (G) with inductived from X topology is the topological prodThe space X i uct G × Xi and the restriction g = f|X˜ is a function of two variables x î and xi . Since ˜ Hence E = Ei,0 S D(g). Clearly, ˜ is an open subspace of X then D(g) = D(f ) T X. X g is continuous with respect to the second variable. Show that g is quasi-continuous ˆ i then for evwith respect to the first variable. Pick x î ∈ G. Since G is open in X ery k = 1, ..., n, k 6= i there exists an open in Xk set Uk such that for the product ˆk is Baire for every k U of sets Uk we have x î ∈ U ⊆ G. Since G is Baire then U too. Fix xi ∈ Xi . Put h = gxi |U . Since h ∈ Sn−1 (U, Z) then by the Breckenridge and Nishiura result [5] the mapping h is quasi-continuous. Thus the mapping gxi is quasi-continuous at x î . Hence g is quasi-continuous with respect to the first variable. Therefore g ∈ KC(G × Xi , Z). By Theorem 2.14 there exists a sequence of locally ˜ = G × Xi such projectively meager with respect to the second axis Fσ -sets Ei,m in X ∞ S ˜ is an open subspace of X then it is an Fσ -set in X. that D(g) = Ei,m . Since X m=1

Then Ei,m are locally projectively meager with respect to the i-th axis Fσ -sets in X. ∞ S Therefore the representation E = Ei,m is to be found. (i)

Now put Em =

m S j=0

m=0

Ei,j and Em =

meager Fσ -sets in X and E =

∞ S m=0

n T i=1

(i)

Em . Clearly Em are locally projectively

Em .

Theorems 2.13 and 2.15 imply the final result on description of discontinuity points set of separately continuous functions on finite products of metrizable spaces. Theorem 2.16. Let X1 , ..., Xn be metrizable spaces, X = X1 × · · · × Xn be the topological product and E ⊆ X. The set E is the discontinuity points set of some separately continuous function f : X → R if and only if E is the union of a sequence of locally projectively meager Fσ -sets.

155


2.3. The exact inverse problem. Solving general inverse problem we was carrying only with existence of discontinuity but not with their size. Therefore it is natural to consider exact inverse problem about construction of separately continuous function f on the product X = X1 , ..., Xn of topological spaces oscillation of which is equal to a given upper semicontinuous function g : X → [0, +∞]. We shall be solving this problem. The given below results are included into [24] in the case n = 2. Let X be a topological space. Denote by Ux the system of all neighborhoods of a point x in X. For f : X → R put f ∗ (x) = inf U ∈Ux sup f (U ) and f∗ (x) = supU ∈Ux inf f (U ). Clearly f∗ (x) ≤ f (x) ≤ f ∗ (x). Functions f ∗ : X → (−∞, +∞] and f∗ : X → [−∞, +∞) are upper and lower Baire functions of function f respectively. They are upper and lower semicontinuous respectively. Besides ωf = f ∗ − f∗ . If g : X → R is continuous at a point x then (f +g)∗ (x) = f ∗ (x)+g(x) and (f +g)∗ (x) = f∗ (x) + g(x). Thus ωf +g (x) = ωf . Evidently f ≤ g implies f ∗ ≤ g ∗ and f∗ ≤ f∗ . Lemma 2.17. Let X be a completely regular space, ϕ : X → [0, +∞) be a continuous function, η > 0, ψ : X → [0, η] be arbitrary function, U0 be a neighborhood of a point p ∈ X and ε > 0, besides ϕ(x) + ψ(x) ≤ η + ε on U0 , (ϕ + ψ)∗ (p) ≥ η − ε and suppϕ ⊆ U0 . Then there exists a continuous function ϕ˜ : X → [0, +∞) such that suppϕ˜ ⊆ U0 , ϕ˜ + ψ ≤ η on U0 , (ϕ˜ + ψ)∗ (p) = η and |ϕ(x) ˜ − ϕ(x)| ≤ ε on X. Proof. Put η0 = (ϕ + ψ)∗ (p) and δ = η + ε − η0 . Since ϕ(x) + ψ(x) ≤ η + ε on U0 , then η0 ≤ η + ε. Thus δ ≥ 0. Consider a decreasing sequence of neighborhoods Un ⊆ U0 of p such that ϕ(x)+ψ(x) ≤ η0 +2−n δ on Un and a corresponding sequence of continuous functions ϕnP: X → [0, 2−n δ] for which suppϕn ⊆ Un and ϕn (p) = 2−n δ. ∞ The function σ(x) = n=1 ϕn (x) is defined and continuous on X as the sum of uniformly convergent series of continuous functions. The continuous function ϕ(x) ˜ = max{0, ϕ(x) + σ(x) − ε} is to be found. The following lemma is the main technical tool in our approach to solving of the exact inverse problem. Lemma 2.18. Let X be completely regular space with the topology T , (Pn )∞ n=0 be a sequence of subsets of X and τn : Pn → T be a disjoint open family such that p ∈ τn (p) for all p ∈ Pn . Let hn : Pn → In be a function, where I0 = [1, +∞) and In = [4−n , 41−n ] if n > 0. Then there exists sequence of families (ϕnp : p ∈ Pn ) of continuous functions ϕnp : X → [0, +∞) such that: (i) suppϕnp ⊆ τn (p) for all n and p ∈ Pn ; P∞ P (ii) the functions fn = p∈Pn ϕnp and rn = j=n fj for x ∈ τn (p) n = 0, 1, ... and p P ∈ Pn satisfy the inequalities rn (x) ≤ hn (p) and rn∗ (p) = hn (p). Besides the series fn is uniformly convergent on X. Proof. At first we shall construct sequence of functional matrixes (ϕnp,k : p ∈ Pn , k = 0, 1, ...) of continuous functions ϕnp,k : X → [0, +∞) such that suppϕnp,k ⊆ τn (p) k P n P ϕp,k and rn,k = for all n, k = 0, 1, ... and p ∈ Pn and functions fn,k = fj,k p∈Pn

j=n

satisfy the inequalities |fn,k+1 − fn,k | ≤ 2−k for all n and k, rn,k (x) ≤ hn (p) if ∗ (p) = hn (p) for all n ≤ k and p ∈ Pn . x ∈ τn (p) and rn,k For any n and p ∈ Pn choose a continuous function ϕnp,0 : X → [0, hn (p)] such that suppϕnp,0 ⊆ τn (p) and ϕnp,0 (p) = hn (p). Suppose that for some k > 0 the functions ϕnp,0 ,...,ϕnp,k−1 are constructed for any n and p ∈ Pn . We shall begin construction of the functions ϕnp,k . If n ≥ k and p ∈ Pn then we put ϕnp,k = ϕnp,k−1 . Suppose that 0 ≤ m < k and the functions ϕnp,k have been constructed for n > m, besides the inequality |fn,k − fn,k−1 | ≤ 21−k−n holds. Since 0 ≤ fk,k = fk,0 ≤ 41−k then

156 V.K.MASLYUCHENKO, O.V.MASLYUCHENKO, V.V.MYKHAYLYUK, O.V.SOBCHUK |fm,k−1 +rm+1,k −rm,k−1 | = | k−1 P

k−1 P

(fj,k −fj,k−1 )+fk,k | ≤

j=m+1

k−1 P j=m+1

|fj,k −fj,k−1 |+fk,k ≤

21−k−j + 41−k−m = 21−k−m .

j=m+1

Pick p ∈ Pn and x ∈ τm (p). Clearly that fm,k−1 (x) = ϕm p,k−1 (x). By the inductive assumption rm,k−1 (x) ≤ hm (p). Hence ϕm p,k−1 (x) + rm+1,k (x) = fm,k+1 (x) + ∗ rm+1,k (x) ≤ rm,k−1 (x) + 21−k−m ≤ hm (p) + 21−k−m and (ϕm p,k−1 + rm+1,k ) (p) = ∗ (fm,k−1 + rm+1,k )∗ (p) ≥ rm,k−1 (p) − 21−k−m = hm (p) − 21−k−m . Show that rm+1,k ≤ −m hm (p). Indeed, 4 ≤ hm (p) since p ∈ Pn . Besides for any n > m and q ∈ Pn the Sn−1 S inequality hn (q) ≤ 41−n ≤ 4−m ≤ hm (p) holds. Thus, if ξ ∈ τn (q) \ j=m+1 τj (Pj ) for some n with m < n ≤ k and q ∈ Pn then rm+1,k (ξ) = rn,k (ξ) ≤ hn (q) ≤ hm (p). In other case rm+1,k (ξ) = 0. Thus the functions ϕ = ϕm p,k−1 , ψ = rm+1,k and the numbers η = hm (p), 1−k−m ε = 2 satisfy the condition of the previous lemma with U0 = τm (p). It im˜ Then suppϕm plies the existence of the function ϕ. ˜ Put ϕm p,m = ϕ. p,k ⊆ τm (p), m rm,k (x) = fm,k (x) + rm+1,k (x) = ϕp,k (x) + rm+1,k (x) ≤ hm (p), for x ∈ τm (p), ∗ ∗ m m 1−k−m rm,k (p) = (ϕm . Hence p,k + rm+1,k ) (p) = hm (p) and |ϕp,k − ϕp,k−1 | ≤ 2 1−k−m 1−k |fm,k − fm,k−1 | ≤ 2 ≤2 . We shall construct the functions ϕnp . Notice, |ϕnp,k+1 −ϕnpk | ≤ |fn,k+1 −fnk | ≤ 2−k for any numbers n, k and poins p ∈ Pn . Therefore the sequence (ϕnp,k )∞ k=0 is uniformly n n convergent to some continuous function ϕ : X → [0, +∞] with suppϕ p p ⊆ τn (p). The P n ϕp are correctly defined on X, besides fn,k → fn for k → ∞. Note functions fn = p∈Pn

that |fn,k − fk | ≤ 21−k , since |fn,k − fn,k+m | ≤

k+m P

|fn,j+1 − fnj | ≤

k+m P

21−j =

j=k+1

j=k+1

21−k . Since fn,k ≤ rn,k ≤ 41−n P for 1 ≤ n ≤ k, then with k → ∞ we have fn ≤ 41−n for any n. Therefore, the series fn is uniformly convergent on X and the functions ∞ P rn = fj are correctly defined on X. Show that rn,k are uniformly convergent to j=n

rn for k → ∞. Indeed, for k ≥ n we have |rn,k − rn | ≤ |rn,k − k P j=n

|fj,k − fj | +

∞ P j=k+1

fj ≤

k P

21−k +

∞ P

k P j=n

fj | + |

k P j=n

fj − r n | ≤

21−j ≤ (k + 2)21−k → 0 for k → ∞.

j=n

j=k+1 ∗ x ∈ τn (p) and rn,k (p) = hn (p) for n for x ∈ τn (p) and rn∗ (p) = hn (p) for

Clearly rn,k (x) ≤ hn (p) for ≤ k and p ∈ Pn . If k → ∞ then rn (x) ≤ hn (p) any numbers n and points p ∈ Pn . The main result we state in more general form. Theorem 2.19. Let X be the product of some completely regular spaces and g : X → [0; +∞] be an upper semicontinuous function such that the sets g −1 ([ε, +∞]) are oblique approximating for all ε > 0 but one of them is strongly oblique approximating (for example if it is a Gδ -set). Then there exists a separately continuous function f : X → [0, +∞) such that ωf = g. Proof. Since ωλf = λωf for λ > 0 then without loss of generality it is possible to assume that S0 = g −1 ([1, +∞]) is strongly oblique approximating. Put S I0 = [1, +∞], n In = [4−n , 41−n ] for n > 0, Sn = g −1 (In ) and Sn0 = g −1 ([4−n , +∞]) = j=0 Sj . The 0 0 sets Sn are oblique approximating and S0 = S0 is strongly oblique approximating. Hence there exist corresponding families πn0 : Sn0 → 2X and τn0 : Pn0 → Bm , where


157

S Pn0 = τn0 (Sn0 ). Besides for n = 0 there exists ν : Pn → N such that for all s ∈ S0 , neighborhood U of s and m ∈ N there exists p ∈ P00 ∪ U such that ν(p) > m. Put S 0 0 πn = πn |Sn and τn = τn |Pn , where Pn = πn (Sn ) for every n. Put Sn (p) = {s ∈ Sn : p ∈ πn (s)} and hn (p) = sup g(Sn (p)) for n > 0 and h0 (p) = min{ν(p), sup g(S0 (p))} for any point p ∈ Pn . By Lemma 2.18, there exists a sequence of families (ϕnp : p ∈ Pn ) n of continuous functions ϕnp : X → [0, +∞) for (p) for each n ∈ N P whichnsuppϕp ⊆ τnP ∞ and p ∈ Pn . Besides, the functions fn = p∈Pn ϕp and rn = j=n fj satisfy P the ∗ following condition: rn (x) ≤ hn (p) for x ∈ τ (p), r (p) = h (p), and the series fn is n n n P∞ uniformly convergent. Put f = n=0 fn . By the construction for every n the system {suppϕnp : p ∈ Pn } is crosswise locally finite. Then the functions fn are separately continuous. It is implies that f is separately continuous too. It possible to show that ωf = g. Proceed to a characterization of oscillation of separately continuous functions. Theorem 2.20. Let X = X1 × · · · × Xn be the product of metrizable spaces such that the product X1 × · · · × Xn−1 is Baire, g : X → [0, +∞] and Sε = g −1 ([ε, +∞]). For the existence of a separately continuous function f : X → R such that ωf = g it is necessary and sufficient that g be upper semicontinuous and for every ε > 0 the function g satisfies one of the following conditions: (i) Sε nowhere dense and covered by countable family of locally projectively meager sets; (ii) Sε nowhere dense and covered by countable family of locally projectively nowhere dense sets; (iii) Sε has a nowhere dense cross-neighborhood; (iv) Sε is oblique approximating. Proof. Clearly, (i) ⇔ (ii). The implications (i) ⇒ (ii) ⇒ (iv) follow from the closeness of Sε and theorems 2.4, 2.5 If g is upper semicontinuous and (iv) holds for all ε > 0 then by theorem 2.19 there exists separately continuous function f : X → R with ωf = g since X is perfect and completely regular. Thus, sufficiency holds even for any metrizable spaces. If g = ωf for some separately continuous function f then g is upper semicontinuous. Besides, the function f is quasi-continuous [5, Theorem 2.2]. Thus for any ε > 0 the set Sε is nowhere dense and Sε ⊆ D(f ). Then (ii) holds for any ε > 0 by Theorem 2.15. If the product X is Baire then we shall be able to obtain more simple characterization of oscillation of separately continuous function. Theorem 2.21. If Baire space X is the product of some metrizable spaces then for the existence of a separately continuous function f : X → R with ωf = g it is necessary and sufficient that g to be upper semicontinuous and its support be covered by the countable family of locally projectively meager sets. Notice, if X is separable then the condition on support become more simply. Namely suppg must be projectively meager set. If X is hereditarily Baire space then we shall be able to obtain other interesting characterization of oscillation of separately continuous functions. Recall that a topological space is called hereditarily Baire if every its closed subspace is Baire. In particular, any completely metrizable space is hereditarily Baire. Theorem 2.22. Let a hereditarily Baire (in particular, completely metrizable) space X be the product of some metrizable spaces and g : X → [0, +∞]. Then for the existence of a separately continuous function f : X → R with ωf = g it is necessary and sufficient that g be upper semicontinuous and each nonempty closed subset of its support be projectively nowhere dense at least at one of its points.

158 V.K.MASLYUCHENKO, O.V.MASLYUCHENKO, V.V.MYKHAYLYUK, O.V.SOBCHUK ˆ i = Q Xk are meager it is possible to obtain the In the case when all products X k6=i

following result. Theorem Q 2.23. Let X be the product of metrizable spaces X1 , ... Xd and the spaces Yi = j6=i Xj be meager for all i. Let g : X → [0, +∞] be upper semicontinuous function such that g −1 ([ε, +∞)) is nowhere dense for any ε > 0. Then there exists a separately continuous function f : X → R with ωf = g. 3. Baire and Lebesgue classification of separately continuous mappings and their analogous. 3.1. The modification of Rudin–Saint-Raymond methods and applications of the extension theorems. Rene Baire [2] give a classification method for real functions which is called Baire classification. It will be able to extend on mappings of general topological spaces. Let X and Y be topological spaces. We call continuous mappings f : X → Y by Baire class zero mappings. The set of this mappings is denoted by C(X, Y ) or B0 (X, Y ). Assume that α > 0 is some finite or countable ordinal and classes Bξ (X, Y ) are defined for all ordinals ξ < α. A mapping f : X → Y is called the Baire class α mapping if there exist a sequence of ordinals ξn < α and a sequence of mappings fn : X → Y such that fn ∈ Bξn (X, Y ) and fn (x) → f (x) on X. By Bα (X, Y ) we denote the family of all these mappings. Clearly, Bξ (X, Y ) ⊆ Bη (X, Y ) for 0 ≤ ξ ≤ η < ω1 . Sometimes it is used another therminology. The Baire class α ˜α (X, Y ) = Bα (X, Y ) \ ( S Bξ (X, Y )). We need following simple facts means a set B ξ 0. Put fn (x, y) =

X

ϕn,i (x)f (xn,i , y)

i∈In

for n ∈ N and (x, y) ∈ X × Y . By the choice of points xn,i we have f xn,i ∈ Bα (Y, Z) for every pair (n, i) ∈ N × In . Then by Lemma 3.2 fn ∈ Bα (X × Y, Z) for every n. Show that fn (x, y) → f (x, y) on X × Y . Fix a point (x0 , y0 ) ∈ X × Y and consider


159

arbitrary convex neighborhood of zero W in Z. Clearly X fn (x0 , y0 ) − f (x0 , y0 ) = ϕn,i (x0 ) (f (xn,i , y0 ) − f (x0 , y0 )) . i∈In

Since the mapping fy0 is continuous then there is the number n0 such that f (x, y0 ) − f (x0 , y0 ) ∈ W if d(x, x0 ) < 1/n0 . Clearly, In,0 = {i ∈ In : ϕn,i (x0 ) 6= 0} is finite. If i ∈ In,0 then d(xn,i , x0 ) < 1/n since the points xn,i and x0 belongs to the support of ϕn,i which contains in some ball of Bn . Then for n ≥ n0 we have fn (x0 , y0 ) − f (x0 , y0 ) = =

X i∈In,0

ϕn,i (x0 ) (f (xn,i , y0 ) − f (x0 , y0 )) ∈

X

ϕn,i (x0 )W ⊆ W.

i∈In,0

Hence fn (x0 , y0 ) → f (x0 , y0 ). Thus f ∈ Bα+1 (X × Y, Z). Corollary 3.4. Let X1 , ..., Xn be metrizable spaces, X = X1 × ... × Xn and Z be an locally convex space. Then Sn (X, Z) ⊆ Bn−1 (X, Z). Notice that this corollary holds in the case when one of the spaces X1 , ..., Xn is probably nonmetrizable. As we see locally convexness of Z is essential. But the usage of the Lebesgue method shows that locally convexness of Z is not necessary in the case of X = Rn [36]. It is naturally to study the class of spaces X such that proceeding theorem holds ˇ for any topological vector spaces Z. Cech-Lebesgue dimension [10, p.564] is useful in this research. Let m be nonnegative integer number. It is said that order of system A less than m if every part of A which consists of more then m sets have empty intersection. It ˇ denote by ordA < m. A Tikhonoff space X is called finite Cech-Lebesgue dimension if there is nonnegative integer m such that every finite functional open covering of X admits inscribing of finite functional open covering of X with order less than m. This is noted by dim < m. By the Dowker theorem [10, p.564] if X is a paracompactum and dimX < m then every open covering of X admits inscribing of open covering with order less then m. We obtain following result using method of Theorem 3.3. and the Douker and the Stone theorems. ˇ Theorem 3.5. Let X be a metrizable space with finite Cech-Lebesgue dimension, Y be a topological space, Z be a topological vector space and α < ω1 . Then CB α (X × Y, Z) ⊆ Bα+1 (X × Y, Z). Loosening of metrizability of X in the Rudin theorem is natural. Below we show that sometimes metrizability of X can be changed to σ-metrizability. The Tietse-Uryson theorem [10, p.116], the Dugundji theorem [32] and new functional interpolation technique [17] are used. A topological space X is called σ-metrizable if it is the countable union of closed metrizable subspaces Xn . The class of σ-metrizable spaces includes strict inductive limits of sequences of metrizable locally convex spaces. Besides, every locally convex space X with the weak topology such that the strong conjugate X ∗ is metrizable and separable belongs to this class. Clearly, every σ-metrizable space is S perfect but it need not be normal (for example, the Nemytsky plane). If X = Xn is a σmetrizable regular space and Xn are separable closed metrizable subspaces then X is Lindelöff perfectly normal space. Noticed by L1 the class of all strictly inductive

160 V.K.MASLYUCHENKO, O.V.MASLYUCHENKO, V.V.MYKHAYLYUK, O.V.SOBCHUK limits of sequences of locally convex metrizable separable spaces and by L2 the class of all Hausdorff locally convex spaces with weak topology strong conjugates of which are metrizable and separable. These classes are invariant under the finite products. Besides, every space from these classes is σ-metrizable Lindelöff perfectly normal space. In particular, the space R∞ of finite real sequences, the space K of continuous function with compact support, the Schwarts space D and the spaces lp , 1 < p < ∞, with weak topology are σ-metrizable. Let E ⊆ X and g : E → R be a function. By gχE is denoted a function f : X → R for which f (x) = g(x) on E and f (x) = 0 on X \ E. Lemma 3.6. Let E be a closed Gδ -subset of normal space X, 0 < α < ω1 and g ∈ Bα (E, R). Then f = gχE ∈ Bα (X, R). Lemma 3.7. Let E be a closed Gδ -subset of normal space X, 0 < α < ω1 ,f ∈ Bα (X, R), g = f |E ang g(x) = lim gn (x) on E where gn : E → R are functions less n→∞ than α Baire class. Then there exits a sequence of functions fn : X → R less than α Baire class such that f (x) = lim fn (x) on X and fn |E = gn . n→∞

Proof. Pick gn ∈ Bξn (E, R) where ξn < α. For every gn there exists a function gñ ∈ Bξn (X, R) such that gñ |E = gn . Since f ∈ Bα (X, R) then there is numbers ηn and functions hn such that ηn < α, hn ∈ Bηn (X, R) and hn (x) → f (x) on X. Choose ∞ S an increasing sequence of closed sets Fn such that Fn = X \ E. Then there exist n=1

−1 continuous functions ϕn : X → [0, 1] such that E ⊆ ϕ−1 n (1) and Fn ⊆ ϕn (0). Put fn (x) = ϕn (x)˜ gn (x) + (1 − ϕn (x))hn (x) for x ∈ X. The sequence fn is to be found. Class of σ-metrizable spaces was defined in [32] and it was used for the generalization of the Rudin theorem. Following result is improved variant of main result of this paper. Theorem 3.8. Let X be a σ-metrizable space, Y be a topological space such that the product X × Y be a perfectly normal and ordinal α < ω1 . Then CBα (X × Y, R) ⊆ Bα+1 (X × Y, R). Proof. Pick f ∈ CBα (X ×Y, R). Represent X as the union of increasing sequence of a closed subspaces Xn . Put fn = f |Xn ×Y . Clearly, fn ∈ CBα (Xn × Y, R). Then by Theorem 3.3 fn ∈ Bα+1 (Xn × Y, R) for every n. Clearly, subspaces Pn = Xn × Y of the product P = X × Y are perfectly normal. Since Pn is closed subspace of Pn+1 then it is Gδ -set in Pn+1 . Since f1 ∈ Bα+1 (P1 , R) then there exists a pointwise convergent to f1 sequence of functions g1,k : P1 → R less than α + 1 Baire class. The function f2 : P2 → R is an extension of f1 and f2 ∈ Bα+1 (P2 , R). Therefore by Lemma 3.7 there exists a sequence of functions g2,k : P2 → R less then α + 1 Baire class such that g2,k |P1 = g1,k for all k and lim g2,k (p) = f2 (p) on P2 . And so on. For every n we have obtained sequences k→∞

of functions gn,k : Pn → R less then α + 1 Baire class such that lim gn,k (p) = fn (p) k→∞

on Pn and gn+1,k |Pn = gn,k , k ∈ N. Since the product P is perfectly normal and the subspace Pn is closed in P then every function gn,n can be extended to a function gn : P → R of the same Baire class. In the case α = 0, it is possible to make using the Titze-Uryson theorem. If α > 0 it is made by Lemma 3.6. It easy to see that lim gn (p) = f (p) on P . n→∞

Below we shall be carrying out exchange of the space R in previous problem by arbitrary locally convex space Z. The first method to do it is to use the Dugundji theorem instead of the Titze-Uryson theorem. Lemma 3.9. Let T be a metrizable space, E be a closed subspaces of T , Z be a


161

locally convex space and g ∈ Bα (E, Z), 0 < α < ω1 . Then a mapping f : T → Z such that f (t) = g(t) on E and f (t) = 0 on T \ E belongs to Bα (T, Z). Lemma 3.10. Let T be a metrizable space, E be a closed subspaces of T , Z be a locally convex space, f ∈ Bα (T, Z), 0 < α < ω1 , , (αn )∞ n=1 be a sequence of ordinals αn < α and the restriction g = f |E be the pointwise limit of mappings gn ∈ Bαn (E, Z). Then there exists a pointwise convergent to f sequence of mapping fn ∈ Bαn (T, Z) such that fn |E = gn . An increasing sequence of closed subspaces Xn of a topological space X is called ∞ S T ideal if X = Xn and a subset F ⊆ X is closed if and only if all sets Fn = F Xn n=1

are closed. In other words, X is the direct limit of Xn . If X is the direct limit of sequence of metrizable subspaces then it is called ideally σ-metrizable space. The space R∞ is typical example of such spaces. Proposition 3.11. Let X be topological space, (Xn )∞ n=1 be an ideal sequence in X, Y be a Hausdorff locally compact space and Tn = Xn × Y . Then (Tn )∞ n=1 is an ideal sequence in T = X × Y . Proof. Show that T is the direct limit of Tn . Consider set F ⊆ T such that sets Fn = F ∩ Tn are closed. Verify that F is closed. Consider t0 = (x0 , y0 ) ∈ T \ F . There is a numbern0 such that x0 ∈ Xn0 and t0 ∈ Tn0 . Since L = {x0 } × Y ⊆ Tn0 then the set F ∩ L = F ∩ Tn0 ∩ L = Fn0 ∩ L is closed in L. Since t0 ∈ L \ (F ∩ L) then there exists compact neighborhood V of y0 in Y such that ({x0 } × V ) ∩ F = Ø. Consider closed in X × V sets Fñ = Fn ∩ (X × V ). By the Kuratowski theorem ∞ S [10, p.200] the projections An of sets Fn on X are closed. Put A = An . Notice n=1

A ∩ Xn = An . Since the sequence (Xn ) is ideal then A is closed in X. It is easy to see that x0 6∈ A. Therefore there is a neighborhood U of x0 in X such that U ∩ A = Ø. Thus W = U × V is a neighborhood of t0 in T . It is possible to show that W ∩ F = Ø. Hence F is closed in T . Proposition 3.12. Let (Tn )∞ n=1 be ideal sequence in a topological space T such that Tn are metrizable, Z be a locally convex space, f : T → Z such that fn = f |Tn ∈ Bα (Tn , Z) for each n. Then f ∈ Bα (T, Z). Proof. We shall prove this theorem by induction. If α = 0 then all mappings fn are continuous. Consider a closed in Z set B. Put F = f −1 (B) and Fn = F ∩ Tn . Clearly Fn = fn−1 (B). Thus F is closed because (Tn )∞ n=1 is ideal. Hence f is continuous. Assume that α > 0 and the statement holds for all ξ < α. Pick some sequence of ordinals αk < α, which tends to α if α a limiting ordinal and αk = β if α = β + 1. Since f1 ∈ Bα (T1 , Z) then by Lemma 3.1 there exists a sequence of mappings g1,k ∈ Bαk (T1 , Z) which pointwise converges to f1 on T1 . Since f1 = f2 |T1 then by Lemma 3.13 there exists a sequence of mappings g2,k ∈ Bαk (T2 , Z) such that g2,k |T1 = g1,k and g2,k (t) → f2 (t) on T2 . As so on. Thus we obtain an infinite matrix of mappings gn,k ∈ Bαk (Tn , Z) for which gn+1,k |Tn = gn,k and gn,k (t) → fn (t) on Tn for every n. The conditions gk |Tn = gn,k uniquely define some mapping gk : T → Z. By the induction hypothesis gk ∈ Bαk (T, Z). It easy to see that gk (t) → f (t) on T . Proposition 3.13. Let X be an ideally σ-metrizable space, Y be a metrizable locally compact space and Z be a locally convex space E. Then CBα (X × Y, Z) ⊆ Bα+1 (X × Y, Z) for every α. The second way is an immediately application of Rudin’s method. It’s defined by existence of special sequences of partitions of unit in the space X. With this Y can be random topological space.

162 V.K.MASLYUCHENKO, O.V.MASLYUCHENKO, V.V.MYKHAYLYUK, O.V.SOBCHUK Theorem 3.14. Let X be a topological space with sequences of locally finite partitions of unit (ϕn,i : i ∈ In ) and families (xn,i : i ∈ In ) of points from X such that (i) for every x ∈ X and any neihborhood U of x there exists a number n0 such that condition x ∈ supp ϕn,i implies xn,i ∈ U for every n ≥ n0 and i ∈ In . Then for every topological space Y and any locally convex space Z we have CBα (X × Y, Z) ⊆ Bα+1 (X × Y, Z). Moreover, if the condition (ii) for every x ∈ X there exists m ∈ N such that |{i ∈ In : ϕn,i (x) 6= 0}| ≤ m for all n ∈ N, holds then previous inclusion holds for any topological P vector space Z. Proof. Given f ∈ CBα (X × Y, Z) and fn (x, y) = ϕn,i (x)f (xn,i , y). Clearly i∈In

fn ∈ Bα (X × Y, Z). The condition (i) implies that fn pointwise converges to f on X × Y . Besides in the case when Z is any topological vector space we need to use the condition (ii). Following theorem gives conditions in which previous theorem can be applied. Theorem 3.15. Let X be a topological space which is countable union of increasing sequence of subspaces Xn and d be a metric on X which on each Xn generates its topology, Y be topological space and Z be locally convex space. Then the inclusion CBα (X × Y, Z) ⊆ Bα+1 (X × Y, Z). holds under one of the following conditions: a) the topology generated by d on X weaker then natural; b) X is paracompactum. Proof. a)Given locally a finite partition of unit (ϕn,i : i ∈ In ) in the metric space 1 (X, d) subordinate to the covering of this space by open balls with the radius 2n . The condition a) implies that (ϕn,i : i ∈ In ) is a partition of unit in the space X with natural topology. For every (n, i) ∈ N × In there exists the smallest number k = k(n, i) such that Sn,i = supp ϕn,i ∩ Xk 6= Ø. Choose any point xn,i ∈ Sn,i . Since the metric d generates on every subspace Xn its topology and diam(supp ϕn,i ) → 0 as n → ∞ then it is easy to see that the condition (i) of Theorem 3.14 holds. b) For all numbers n ≤ m consider a covering Bn,m of space Xm by open balls with 1 . For every B ∈ Bn,m there exists an open set U in X such that U ∩Xm = B. radius 2n ∞ S The system of all these sets U denote by Un,m . The system Un = Un,m is an open m=n

covering of X for every n ∈ N. Since X is paracompactum then for every n ∈ N there exists a locally finite partition of unit (ϕn,i : i ∈ In ) subordinate to Un . We choose points xn,i as above. Show that condition (i) holds. Given x ∈ X and U neighborhood of x in X. There exist a number l such that x ∈ Xl and a ball B in Xl with the center at x and the radius δ such that B ⊆ Ul = U ∩ Xl . Choose n0 such that n0 ≥ l and 1 n0 < δ. Suppose that n ≥ n0 , i ∈ In and x ∈ supp ϕn,i . Then (supp ϕn,i ) ∩ Xl 6= Ø, hence k = k(n, i) ≤ l. Therefore xn,i ∈ Xl . There exist m ≥ n and U ∈ Un,m such that supp ϕn,i ⊆ U . Since m ≥ l then (supp ϕn,i ) ∩ Xl ⊆ U ∩ Xm ∈ Bn,m . Then 1 < δ. Hence xn,i ∈ B ⊆ Ul ⊆ U . d(xn,i , x) < m Now we generalize Proposition 3.13 using the Hausdorff theorem on extension of metric [10, p.439]. Corollary 3.16. Let X be an σ-metrizable paracompactum, Y be a topological space and Z be a locally convex space. Then CBα (X × Y, Z) ⊆ Bα+1 (X × Y, Z).


163

Since ideal σ-metrizable space is stratifiable, hence it is paracompact [4], then this Corollary implies that Proposition 3.13 holds for any topological space Y . Clearly that the class of Lindelöff σ-metrizable spaces is invariant uder finite products and besides spaces from this class are paracompactums, hence following statement holds. Corollary 3.17. Let X1 , ..., Xn−1 be Lindel¨ off σ-metrizable spaces, Xn be a topological space and Z be a locally convex space. Then Sn (X1 × · · · × Xn , Z) ⊆ Bn−1 (X1 × · · · × Xn , Z). In the case when X is a topological vector space the metric d is constructed immediately. Corollary 3.18. Let X be a topological vector space which is the countable union of increasing sequence of metrizable linear subspaces Xn , Y be a topological space and Z be a locally convex space. Then CBα (X × Y, Z) ⊆ Bα+1 (X × Y, Z). These statements do not give all examples of spaces to which Theorem 3.14 can be applied. In this connection, T.Banakh [3] has proposed to consider class of spaces X which admit sequence of open coverings Un and functions sn : Un → X such that for any x ∈ X and x ∈ Un ∈ Un , n ∈ N, the sequence of sn (Un ) converges to x. This class is rather wide. It includes so-called semi-stratifable spaces. In particular, spaces with σ-discrete network are the same. It is easy to see that spaces from this class satisfy the conditions of the first part of Theorem 3.14 if, besides, they are paracomtpacta. But the Nemytsky plane satisfies conditions of Theorem 3.14 and is not paracompactum. 3.2. Functions with given diagonal. For any set X and integer n ≥ 2 consider the diagonal map dn : X → X n , dn (x) = (x, ..., x). The set ∆n = dn (X) is called the diagonal of the product X n and the composition g = f ◦ dn : X → Y is called the diagonal of a mapping f : X n → Y . By corollaries 3.4 and 3.17 the inclusion Sn (X n , R) ⊆ Bn−1 (X n , R) holds in the case of metrizable space X or in the case when X belongs to the class Li . It implies that in these cases the diagonal g = f ◦ dn : X → R of every separately continuous function f : X n → R belongs to (n − 1)-th Baire class. We specify conditions on X at which for every function g ∈ Bn−1 (X, R) there exists a function f ∈ Sn (X n , R) such that g = f ◦ dn . In the case X = R this problem was solved by R.Baire [2] and V.Volterra for n = 2 and by H.Lebesgue [16] for n ≥ 2. Their approach used notions of the Euklid geometry essentially. In papers [30,31,21] it was obtained more general results, in which the initial Lebesgue idea has received the natural topological form. We begin from the case n = 2. Lemma 3.19. Let F be a closed set, Wk be open subsets of a normal space T such that F ⊆ Wk for every k ∈ N. Then there exist a sequence of open in T sets Gk and a sequence of continuous functions ϕk : T → [0, 1] such that G0 = T , ∞ P ϕk (t) = 1 F ⊆ Gk+1 ⊆ Gk ⊆ Wk and suppϕk ⊆ Gk−1 \ Gk+2 for k ∈ N. Besides on H = T \

∞ T k=1

Gk = T \

∞ T k=1

k=1

Gk .

164 V.K.MASLYUCHENKO, O.V.MASLYUCHENKO, V.V.MYKHAYLYUK, O.V.SOBCHUK Theorem 3.20. Let X be a Gδ -diagonal space with normal square X 2 , g : X → R be a Baire one function. Then there exists a separately continuous function f : X 2 → R such that f (t) = g(t) on ∆2 . Proof. Consider a pointwise convergent to g sequence of continuous functions gk : X 2 → R. For every t = (x, x) ∈ ∆2 and k find a open neighborhood Uk (x) of x in X such that the oscillation ωgk+1 (Vk (t)) of gk+1Son the neighborhood Vk (t) = Uk (x) × Uk (x) of t in X 2 less then k1 . Put Vk = Vk (t). Clearly that sets Vk 2

t∈∆2

are open. Besides, if p = (x, y) ∈ Vk then inequalities |gk+1 (p) − gk+1 (a)| ≤ k1 and |gk+1 (p) − gk+1 (b)| ≤ k1 hold for points a = (x, x) and b = (y, y). ∞ T Given a sequence of open sets Uk in X 2 such that Uk = ∆2 . Since X 2 k=1

is normalTthen ∆2 is closed in X 2 . By Lemma 3.19 with T = X 2 , F = ∆2 and Wk = Uk Vk we obtain needed sequences of sets Gk and functions ϕk . Then function ∞ P f = g · χ∆2 + ϕk gk is to be found. k=1

Since each first Baire class function g : X → R is the diagonal of some first Baire class function g˜ : X 2 → R (for example, g˜(x, y) = g(x)) then we obtain following corollary. Corollary 3.21. Let X be a Gδ -diagonal space with normal square X 2 . Then every first Baire class function g : X → R is the diagonal of some separately continuous function f : X 2 → R. Now we pass to considering of the general case. We use following notation. For every point x = (x1 , ..., xn ) ∈ X n and index i = 1, ..., n put x î = (x1 , ..., xi−1 , xi+1 , ..., xn ), qi (x) = x î and Di = qi−1 (∆n−1 ). Sometimes we use more exact notation qin and Din instead of qi and Di . The mapping qin : X n → X n−1 is called by projection parallel to i-th The set Din is called byi-th bisector in X n . Clearly qin (∆n ) = ∆n−1 and T axis. n n Di Dj = ∆n for i 6= j. First of all, we shall be studying condition implying that diagonal ∆n of X n has type Gδ . A set A of topological spaces T is called by the set ot type Gδ if there exists ∞ T a sequence of open subsets Gk of T such that Gk+1 ⊆ Gk for all k and A = Gk . k=1

Lemma 3.22. Let X be a topological space such that X 2 is normal. Then following conditions are equivalent. (i) the diagonal ∆n has type Gδ in X n for all n; (ii) the diagonal ∆n has type Gδ in X n for some n; (iii) the diagonal ∆2 has type Gδ in X 2 ; (iv) the diagonal ∆2 is functionally closed in X 2 ; (v) the diagonal ∆n is functionally closed in X n for all n; (vi) the bisectors Din is functionally closed in X n for all n; (vii) the bisectors Din is functionally closed in X n for some i and n; (viii) the diagonal ∆n has type Gδ in X n . Following lemma is used for the inductive step from n − 1 to n in the proof of the main theorem. Lemma 3.23. Let X be a Gδ -diagonal space with normal square X 2 . Then for all n ≥ 3 and i = 1, ..., n there exists a continuous on the set X n \ ∆n function ψi : X n → [0, 1] which equals to 1 on Din \ ∆n and to 0 on Djn for j 6= i. Proof. By Lemma 3.22 all bisectors Djn are functionally closed in X n . Thus there S n are continuous functions ψ˜j : X n → [0, 1] such that Djn = ψ˜j−1 (0). Put Di∗ = Dj j6=i

165


and ψi∗ =

Q ˜ ψj . Obviously ψi∗ is continuous and (ψi∗ )−1 (0) = Di∗ . Functions ψi are

j6=i

defined by  

ψi∗ (x) , if x ∈ X n \ ∆n , ψi (x) = ψ˜i (x) + ψi∗ (x)  0, if x ∈ ∆n . In the proof of following theorem we denote by ϕ · χA a function on T which equals to ϕ : A → R on A ⊆ T and to 0 outside A. Theorem 3.24. Let X be a Gδ -diagonal space with normal square X 2 . Let g : X → R be a Baire (n−1) class function. Then there exists a separately continuous function f : X n → R the diagonal of which equals to g. Proof. Use induction by n. In the case n = 2 the statement of the theorem is identical to Corollary 3.2. Suppose that for n ≥ 3 the theorem is proved when number of variables equals to n − 1. Consider a pointwise convergent to g sequence of Baire n − 2 class functions gk . By the inductive assumption for every k there exists a separately continuous function fk : X n−1 → R with the diagonal gk . Applying Lemma 3.19 in the case T = X n , F = ∆n and Wk are open sets intersection of which equals to ∆n we obtain appropriate sequences of open sets Gk and functions ∞ ∞ T T Gk = ϕk : X n → [0, 1]. Clearly Gk = ∆n . For every i = 1, ..., n put k=1

k=1

hi = g ◦ d−1 n · χ∆n +

∞ X

ϕk · (fk ◦ qin ).

k=1

Clearly that the open set H = X n \ ∆n is covered by locally finite in H sequence ∞ P of open sets Hk = Gk−1 \ Gk+2 . Besides hi |H = ϕk |H · (fk ◦ qin )|H , the functions k=1

hki = ϕk · (fk ◦ qin ) are separately continuous and supphki ⊆ Hk . Then the function hi is separately continuous on H. Since ϕk |∆n = 0 for every k then hi |∆n = g ◦ d−1 n . Show that hi is continuous with respect to i-th variable on the diagonal ∆n . Given p0 = dn (x0 ) ∈ ∆n and fix ε > 0. Choose k0 such that |gk (x0 ) − g(x0 )| ≤ ε for all k ≥ k0 . Since Gk0 +1 is open and p0 ∈ Gk0 +1 then there exists a neighborhood U of x0 in X such that U n ⊆ Gk0 +1 . Put Ui = {(x1 , ..., xn ) ∈ X n : xi ∈ U and xj = x0 for j 6= i}. Consider p ∈ Ui \ {p0 } and show that |hi (p) − hi (p0 )| ≤ ε. In first qin (p)T = dn−1 (x0 ). Thus (fk ◦ qin )(p) = fk (qin (p)) = fk (dn−1 (x0 )) = gk (x0 ). Besides suppϕk Gk0 +1 = ∅ for k < k0 . Then ϕk (p) = 0 for k < k0 because p ∈ Ui ⊆ U n ⊆ Gk0 +1 . Therefore |hi (p) − hi (p0 )| = |

∞ X

ϕk (p)gk (x0 ) − g(x0 )| = |

k=1

=|

X

ϕk (p)(gk (x0 ) − g(x0 ))| ≤

k≥k0

∞ X

ϕk (p)(gk (x0 ) − g(x0 ))| =

k=1

X k≥k0

ϕk (p)|gk (x0 ) − g(x0 )| ≤ ε

X

ϕk (p) ≤ ε.

k≥k0

Given functions ψi from Lemma 3.23. The function f = g ◦ d−1 n · χ∆n + is to be found.

n P i=1

ψ i hi

166 V.K.MASLYUCHENKO, O.V.MASLYUCHENKO, V.V.MYKHAYLYUK, O.V.SOBCHUK Now we input classes Mα (X × Y, Z) for α < ω1 . Put M0 (X × Y, Z) = CC(X × Y, Z). For α > 0 the class Mα (X ×Y, Z) consists of all pointwise limits of sequences of functions fn ∈ Mξn (X ×Y, Z) where ξn < α. Remark that Mα (X ×Y, Z) ⊂ Bα+1 (X × Y, Z) in some important cases. But there is following problem: if CBα (X × Y, Z) ⊂ Mα (X × Y, Z) at least for X = Y = [0, 1], Z = R, and α = 1? Given a decreasing sequence of subset Gn of topological space X. For every ∞ T x ∈ X\ Gn an integer k = k(x), k(x) = max{n : x ∈ Gn } if x ∈ G1 and k(x) = 1 n=1

if x 6∈ G1 is called {the number of the point x generated by the sequence (Gn ). If Gn+1 ⊆ Gn , n = 1, 2,..., then a sequence of continuous functions ϕn : X → [0, 1] −1 which satisfies the condition Gn+1 ⊆ ϕ−1 n (0) and X\Gn ⊆ ϕn (1) for every n ∈ N is called the sequence agreed with the sequence (Gn ). Lemma 3.25. Let X be a normal space and (Gn ) be a decreasing sequence of open sets in X such that Gn+1 ⊆ Gn , n = 1, 2,.... Then there exists a sequence of functions agreed with (Gn ). Two multiindices a = (α1 , ..., αl ) and b = (β1 , ..., βl ) are called symmetrical (a ∼ b) if one of them is obtained by a permutation of coordinates another. A family (ga : a ∈ Nl ) of functions is called symmetrical with respect to multindices if a ∼ b implies ga = gb . Lemma 3.26. Let X be a Gδ -diagonal space with normal square X 2 . Let (gα1 ,...,αm : α1 , ..., αm ∈ N) be a family of continuous functions. Then there exists a sequence of open set Gn in X 2 for which ∞ T (i) Gn = ∆2 ; n=1

(ii) Gn+1 ⊆ Gn for all n ∈ N; (iii) |gα1 ,...,αm (x) − gα1 ,...,αm (y)| < (x, y) ∈ Gn .

1 for every n ∈ N, α1 , ..., αm ≤ n and n

Proof. Given a decreasing sequence (Vn ) of open in X 2 sets such that ∆2 =

∞ T n=1

Vn .

Since gα1 ,...,αm are continuous then for every x ∈ X and n ∈ N there exists an open neighborhood Uxn of x in X such that the of gα1 ,...,αm on Uxn less then S oscillation 1 n n (Ux × Ux ) and Wn = Hn ∩ Vn . Put n for all α1 , ..., αm ≤ n. Put Hn = x∈X

G1 = W1 . Since ∆2 ⊆ G1 ∩ W2 then there exists an open set G2 in X 2 such that ∆2 ⊆ G2 ⊆ G2 ⊆ G1 ∩ W2 . And so on. Thus we have obtained a sequence of sets Gn for which Gn+1 ⊆ Gn ⊆ Gn ⊆ Wn for all n ∈ N. Lemma 3.27. Let (ga : a ∈ Nl ) be a symmetrical family of Baire (k + 1) class functions on topological space X. Then there exists a symmetrical family (ga,αl+1 : a ∈ Nl , αl+1 ∈ N) of Baire k class functions such that lim ga,αl+1 (x) = ga (x) for αl+1 →∞

every a ∈ Nl and x ∈ X. Proof. Put |a| = α1 + ... + αl . Clearly that there is a representation Nl = {a1 , a2 , ...} such that |a1 | ≤ |a2 | ≤ .... Consider for every n ∈ N a pointwise convergent to gan sequence (han ,αl+1 )∞ αl+1 =1 of Baire k class functions. The functions gan ,αl+1 for all αl+1 ∈ N will be defined by recursion on n. If there exist m < n and β ∈ N such that (an , αl+1 ) ∼ (am , β) then gan ,αl+1 = gam ,β else gan ,αl+1 = han ,αl+1 . The symmetrical family (ga,αl+1 : a ∈ Nl , αl+1 ∈ N) is to be found. Theorem 3.28. Let X be a Gδ -diagonal space with normal square X 2 and g : X → R be a Baire (m + 1) class function. Then there exists a function f ∈ CBm (X × X, R) ∩ Mm (X × X, R) such that f (x, x) = g(x) for every x ∈ X.


167

Proof. Consider a pointwise convergent to g on X sequence (gα1 )∞ α1 =1 of Baire m class functions. Applying sequentially m times Lemma 3.29 we obtain symmetrical with respect to multiindices families (gα1 ,...,αk+1 : α1 , ..., αk+1 ∈ N) of continuous functions such that lim gα1 ,...,αk+1 (x) = gα1 ,...,αk (x) for every α1 , ..., αk+1 ∈ N, αk+1 →∞

1 ≤ k ≤ m and x ∈ X. Using Lemma 3.26 to the family (gα1 ,...,αm+1 : α1 , ..., αm+1 ∈ N) we obtain a sequence (Gn ) of open sets Gn ⊆ X 2 . By Lemma 3.25 choose an agreed with (Gn ) sequence of continuous functions ϕn : X 2 → [0, 1]. Put for every multiindex (α1 , ..., αm ) ½ ϕk (x, y)gα1 ,...,αm ,k (x) + (1 − ϕk (x, y))gα1 ,...,αm ,k+1 (x), x 6= y fα1 ,...,αm (x, y) = gα1 ,...,αm (x), x = y where k = k((x, y)) is the number of (x, y) generated by (Gn ). These functions are separately continuous. As αm → ∞, ..., α1 → ∞ we obtain a function ½ ϕk (x, y)gk (x) + (1 − ϕk (x, y))gk+1 (x), if (x, y) ∈ X 2 \∆2 f (x, y) = g(x), if(x, y) ∈ ∆2 which is Baire m class with respect to each variables. The continuity of f with respect to the second variable is implied from gk → g as k → ∞. 3.3. Paracompactness and Lebesgue classification. A mapping f from a perfect space X to a topological space Y is called by Lebesgues α class mapping if for every closed subset B of Y the inverse image f −1 (B) is the multiplicative α class subset of X [10, p.99-100]. The set of all such mappings is denoted by Hα (X, Y ). In 1931 C.Kuratowski [20] proved that CHα (X × Y, Z) ⊆ Hα+1 (X × Y, Z) where X be a metrizable and separable space, Y and Z be a metrizable. This result was generalized by M.Montgomery [34] and C.Kuratowski [21] on the case non separable X. Our attitude to this problem is founded on Stone theorem. Our method need one modification of the Michael theorem on locally finite unions of α class sets. In first, we prove one simple result. Lemma 3.29. Let α < ω1 be the limit of a decreasing sequence of ordinals αn < α and A be a multiplicative (resp. additive) α class subset of a perfect space X. Then there exists a decreasing (resp. increasing) sequence of multiplicative (resp. additive) ∞ ∞ T S αn class sets An such that A = An (A = An ). n=1

n=1

Theorem 3.30. Let A be the union of a family (Ai : i ∈ I) of multiplicative (resp. additive) α class subsets of a perfect space X and (VT s : s ∈ S) be an open locally finite covering of X such that the set Is = {i ∈ I : Ai V s 6= Ø} is finife for all s ∈ S. Then A is multiplicative (resp. additive) α class set. Proof. Use the transfinite induction on α. In the case α = 0 the theorem is obvious. Given 0 < α < ω1 . If α is limit ordinal then choose a increasing sequence of ordinals αn < α tending to α. If α = β + 1 then put αn = β for all n. Assume that our theorem is true for each αn . Prove it for α. Assume Ai ∈ Gα for all i ∈ I. By Lemma 3.29 for each i ∈ I there is an ∞ S increasing sequence of sets Ai,n ∈ Fαn with Ai = Ai,n . Since Ai,n ⊆ Ai , then n=1 T sets Is,n = {i ∈ I : Ai,n V s 6= Ø} are finite. Then by inductive assumption ∞ S S Bn = Ai,n ∈ Fα,n for all n. Since A = Bn then A ∈ Gα . i∈I

n=1

168 V.K.MASLYUCHENKO, O.V.MASLYUCHENKO, V.V.MYKHAYLYUK, O.V.SOBCHUK Assume Ai ∈ Fα for all i ∈ I. By Lemma 3.29 for each i ∈ I there exists ∞ T a decreasing sequence of sets Ai,n ∈ Gαn such that Ai = Ai,n . Consider sets n=1 T S Si = {s ∈ S : Ai V s = Ø}, Fi = V s and Gi = X \ Fi for each i ∈ I. Clearly s∈Si T Ai F Ti = Ø. Hense Ai ⊆ Gi . The sets Fi are T closed and the sets GiTare open. If Ai VTs = Ø then s ∈ Si , V s ⊆ FiTand Gi V s = Ø. Besides if Gi V s = Ø then Ai VT s = Ø. Thus {i S∈ I : Gi V s 6= Ø} = Is is finite for all s ∈ S. Put Bi,n = Ai,n Gi and Bn = Bi,n . Since the set Gi is open then Bi,n ∈ Gαn for each i∈I T i ∈ I. Clearly Bi,n ⊆ Gi for all n ∈ N. Then Is,n = {i ∈ I : Bi,n V s 6= Ø} ⊆ Is is finite for all s ∈ S and n ∈ N. By inductive assumption we obtain Bn ∈ Gαn for all ∞ T n. It is possible to show that Bn = A. n=1

Previous theorem implies following corollaries. Corollary 3.31. Let X be a paracompact perfect space and A be the union of a locally finite family (Ai : i ∈ I) of multiplicative (resp. additive) α class subsets of X. Then A has the same class. Corollary 3.32. Let X be a metrizable space, Y be a perfect space, (Ai : i ∈ I) be a locally finite family of multiplicative (resp. additive) α class subsets of X and S (Ai × Bi ) has (Bi : i ∈ I) be a family of the same class subsets of Y . Then E = i∈I

the same class. Let us pass to the our generalization of the Kuratowski-Montgomery theorem. Theorem 3.33. Let X be a metrizable space, Y be a perfect space and Z be a perfectly normal space. Then CH α (X × Y, Z) ⊆ Hα+1 (X × Y, Z). Proof. Fix a metric | · − · | generating the topology of X. Denote by Uε (x) the ε-neighborhood of x. By the Stone theorem for each k there is an open locally finite covering (Vk,s : s ∈ Sk ) of X consisting from nonempty sets and inscribed in a covering 1 1 (x) : x ∈ X) of X. Clearly diamVk,s < (U 3k k for k ∈ N and s ∈ Sk . Consider a mapping f ∈ CH α (X × Y, Z). Since the set XHαT(f ) is dense in X and the sets Vk,s are open and nonempty then choose xk,s ∈ Vk,s XHα (f ) for each k ∈ N and s ∈ Sk . Given a closed subset F of Z. Since Z is perfectly normal then there exists a sequence of open subsets Gm of Z such that Gm+1 ⊆ Gm for all m and ∞ ∞ T T F = Gm . Put Em,k,s = Vk,s × (f xk,s )−1 (Gm ). Show that Gm = m=1

m=1

f −1 (F ) =

∞ [ ∞ [ \

Em,k,s .

m=1 k=m s∈Sk

Given p = (x, y) ∈ f −1 (F ). Then f (p) ∈ F . Thus f (p) ∈ Gm for every m. Fix m. Since fy (x) = f (x, y) ∈ Gm then x ∈ fy−1 (Gm ). Continuity of the mapping fy : X → Z implies the openness of Hm = fy−1 (Gm ). Hence there is k ≥ m with U k1 (x) ⊆ Hm . Since the sets Vk,s cover X then there exists s ∈ Sk with x ∈ Vk,s . But diamVk,s < k1 . Thus Vk,s ⊆ U k1 (x). Then Vk,s ⊆ Hm . It implies xk,s ∈ Hm that is f (xk,s , y) ∈ Gm or, in another way, y ∈ (f xk,s )−1 (Gm ). Thus p ∈ Em,k,s . Hence p belong to the left part of the needed equality. Assume that for every m there exist k ≥ m and s ∈ Sk with p = (x, y) ∈ Em,k,s . Given m1 = m. Then there exist k1 ≥ m and s ∈ Sk1 such that p ∈ Em1 ,k1 ,s1 . That is x ∈ Vk1 ,s1 and f (xk1 ,s1 , y) ∈ Gm . For m2 = k1 + 1 choose k2 ≥ m2 and s2 ∈ Sk2 whith p ∈ Em2 ,k2 ,s2 . It implies x ∈ Vk2 ,s2

169


and f (xk2 ,s2 , y) ∈ Gm2 . Clearly k2 > k1 and m2 > m. Then Gm2 ⊆ Gm and f (xk2 ,s2 , y) ∈ Gm . And so on. We have obtained the sequence (km ) and the sequence of sn ∈ Skn such that m ≤ k1 < k2 < ... < kn < ..., x ∈ Vkn ,sn and f (xkn ,sn , y) ∈ Gm for each n. Since |xkn ,sn − x| ≤ diamVkn ,sn < k1n then xkn ,sn → x for n → ∞. Hence by the continuity of fy we conclude that f (xkn ,sn , y) → f (p) for n → ∞. Thus ∞ T f (p) ∈ Gm . It implies f (p) ∈ Gm for every m. Then f (p) ∈ Gm = F . That is m=1

p ∈ f −1 (F ). Notice that Vk,s are open in X and (f xk,s )−1 (Gm ) are additive α class subsets of Y . It is implied from f xk,s ∈ Hα (Y, Z). SinceS(Vk,s : s ∈ Sk ) is locally finite then by Corollary 3.32 for every m and k Em,k = Em,k,s is additive α class set. S

Hence Em =

s∈Sk

Em,k has the same class for every m. Thus f −1 (F ) =

k≥m

∞ T m=1

Em is

multiplicative α + 1 class set. It implies f ∈ Hα+1 (X × Y, Z). Corollary 3.34. Let X be a perfect space, Y be a perfectly normal space and a mapping f∞ : X → Y be the pointwise limit of mappings fn ∈ Hα (X, Y ). Then f∞ ∈ Hα+1 (X, Y ). 3.4. Baire one functions and partitions of unit. R.Baire [2] shown that every real function is Baire one if and only if it is pointwise discontinuous on every closed set. Baire one mappings f : X → Y where X is Baire and Y is metrizable are pointwise discontinuous [6, p.125]. We shall solve the inverse problem. Let X be a topological space and ε > 0 . It is said that a function f : X → R is uniformly ε-approximated by Baire one function if there exists Baire one function g : X → R such that |f (x) − g(x)| ≤ ε for every x ∈ X and it is locally ε-approximated by Baire one function if for every x in X there exists its neighborhood U such that f|U is uniformly ε-approximated by Baire one function. Proposition 3.35. Let X be a paracompactum, ε > 0 and f : X → [0, 1] be a locally ε-approximated by Baire one function mapping. Then f is uniformly ε-approximated by Baire one function mapping. Proof. Given a locally finite cover U = (Ui : i ∈ I) of paracompactum X by open sets Ui such that f |Ui is uniformly ε-approximated by Baire one function. By [43, .447] it is possible to choose a cover U such that there exists a partition of unit (ϕi : i ∈ I) on X for which supp(ϕi ) = Ui . For every i ∈ I pick a sequence of continuous functions fn,i : Ui → [0, 1] such that lim fn,i (x) = fi (x) on Ui . Put fn,i (x) = 0 and fi (x) = 0 n→∞

for x ∈ X \PUi . The functions ϕi fn,i are continuous on X and supp(ϕi fn,i ) ⊆ Ui . ϕi fn,i are continuous too. Therefore g : X → R, Thus gn = i∈I

g(x) = lim gn (x) = n→∞

= lim

X

n→∞

ϕi (x)fn,i (x) =

X

i∈I

ϕi (x)fi (x)

i∈I

is Baire oneP function. Notice Pthat |ϕi (x)f (x) − Pϕi (x)fi (x)| ≤ εϕi (x). Hence |f (x) − g(x)| = = | ϕi (x)f (x) − ϕi (x)fi (x)| ≤ ε ϕi (x) = ε. i∈I

i∈I

i∈I

Theorem 3.36. Let X be a perfect paracompactum, f : X → R be a function which is pointwise discontinuous on every closed in X set. Then f is Baire one.

170 V.K.MASLYUCHENKO, O.V.MASLYUCHENKO, V.V.MYKHAYLYUK, O.V.SOBCHUK Proof. Fix some homeomorphism ϕ : R → (0, 1). Put f˜ = ϕ ◦ f . Clearly f˜ is pointwise discontinuous on every closed set and f is Baire one function if and only if f˜ is Baire one. Fix ε > 0. Given collection G of all open in X sets S G such that f˜ is uniformly ε-approximated by Baire one functions on G. Put G = G. It ieasy to see that G is open and it contains continuity points of f˜. Consider the closed set F = X \ G. Show that F = Ø. Suppose the contrary. Given x0 such that f˜|F is continuous in x0 and put y0 = f˜(x0 ). Since F is closed in the perfectly normal space X then there ∞ T exists a sequence (Gn )∞ Gn = F . n=1 of open sets Gn such that Gn+1 ⊆ Gn and n=1

Put G0 = X. By the Vedenisoff theorem [10, p.82] for every n ∈ N there exists a continuous function ψn : X → [0, 1] such that ψn−1 ((0, 1]) = Gn−1 \ Gn+1 P = Un . Collection (Un : n ∈ N) is a locally finite open covering of G. Thus ψ = ψn n∈N

is positive and continuous on G and a sequence of functions (ϕn = ψψn : n ∈ N) is locally finite partition of unit on G. Besides ϕ−1 n ((0, 1]) = Un . Consider spaces Xn = X \ Gn+1 which are perecompactums as closed subspaces of paracompactum. Since Xn ⊆ G for every n ∈ N then by Proposition 3.35 functions fn = f˜|Xn are uniformly ε-approximate by Baire one function. Hence there exists a sequence of Baire one function gñ : Xn → R such that |gñ (x) − fn (x)| ≤ ε for every x ∈ Xn . For every n ∈ N choose a sequence of bounded continuous functions gn,k : Xn → R such that lim gnk (x) = gñ (x) on Xn . Put gn,k = 0 on G \ Xn and consider k→∞ n ∞ P P continuous functions gn : X → R, gn (x) = ϕk (x)gkn (x) + y0 ϕk (x) if x ∈ G k=1 k=n+1 T T and gn (x) = y0 if x ∈ F . Since (supp(ϕk )) Gk+1 = Ø then (supp(ϕk )) Gn+1 = Ø ∞ P for every k ≤ n. Thus for x ∈ Gn+1 \ F we have gn (x) = y0 ϕk (x) = y0 . k=n+1

Hence gn|Gn+1 ≡ y0 and gn is continuous on F . Passing to the limit as n → ∞ we ∞ P ϕk (x)gk (x) if x ∈ G and g(x) = lim gn (x) = y0 obtain that g(x) = lim gn (x) = n→∞

n→∞

k=1

if x ∈ F . As in the proof of the previous Proposition for x ∈ G we have |g(x) − ∞ ∞ ∞ ∞ P P P P ϕk (x)ε = ϕk (x)(gk (x) − f˜(x))| ≤ ϕk (x)f˜(x)| = | ϕk (x)gk (x) − f˜(x)| = | k=1

k=1

k=1

k=1

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