paper also contains a comparison of cardinal functions on D* p(X) and on real- valued continuous functions Cp(X) and a generalization of a sufficient condition.
DOI: 10.2478/s12175-008-0090-3 Math. Slovaca 58 (2008), No. 4, 509–520
SOME NOTES ON DENSELY CONTINUOUS FORMS ˇ Peter Vadovic (Communicated by L’ubica Hol´ a) ABSTRACT. We consider a special space of set-valued functions (multifunctions), the space of densely continuous forms D(X, Y ) between Hausdorff spaces X and Y , defined in [HAMMER, S. T.—McCOY, R. A.: Spaces of densely continuous forms, Set-Valued Anal. 5 (1997), 247–266] and investigated also in ´ L’.: Spaces of densely continuous forms, USCO and minimal USCO [HOLA, maps, Set-Valued Anal. 11 (2003), 133–151]. We show some of its properties, ´ D.—VADOVIC, ˇ P.: Densely concompleting the results from the papers [HOLY, tinuous forms, pointwise topology and cardinal functions, Czechoslovak Math. J. ´ D.—VADOVIC, ˇ P.: Hausdorff graph topo58(133) (2008), 79–92] and [HOLY, logy, proximal graph topology and the uniform topology for densely continuous forms and minimal USCO maps, Acta Math. Hungar. 116 (2007), 133–144], in particular concerning the structure of the space of real-valued locally bounded densely continuous forms Dp∗ (X) equipped with the topology of pointwise convergence in the product space of all nonempty-compact-valued multifunctions. The paper also contains a comparison of cardinal functions on Dp∗ (X) and on realvalued continuous functions Cp (X) and a generalization of a sufficient condition for the countable cellularity of Dp∗ (X). c 2008 Mathematical Institute Slovak Academy of Sciences
1. Introduction For Hausdorff spaces X, Y , Z, we denote by F (X, Y ) the space of all functions from X to Y ; by 2Z , K (Z) and F (Z) we mean the family of all closed, nonempty compact and finite subsets of Z, respectively. Thus F (X, 2Y ) denotes the space of all closed-valued multifunctions from X to Y . Also, B(x) denotes the local base of open sets at a point x; int A and A denote the interior and closure of A. 2000 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 54C60; Secondary 54A25, 54C35. K e y w o r d s: locally bounded densely continuous form, pointwise convergence, Fr´echet space, Volterra space. This work was supported by Science and Technology Assistance Agency under the contract No. APVT-51-006904 and by the Eco-Net (EGIDE) programme of the Laboratoire de Math´ ematiques de l’Universit´e de Saint-Etienne (LaMUSE), France.
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Now we define the main object of our interest here, the space of densely continuous (or DC) forms D(X, Y ). First, a single-valued function f from X to Y is called densely continuous (or f ∈ DC(X, Y )), if the set C(f ) of points of continuity of f is dense in X. For any f ∈ DC(X, Y ) let φf denote the closure in X × Y of the graph of the restriction f C(f ). The space D(X, Y ) of densely continuous forms then consists of the elements φf for all f ∈ DC(X, Y ). Of course, every densely continuous form has closed graph in X × Y and can be re garded as a closed-valued multifunction, if we put φf (x) = y ∈ Y : (x, y) ∈ φf for each x ∈ X. Thus D(X, Y ) ⊆ F (X, 2Y ). The subspace D∗ (X, Y ) of locally bounded densely continuous forms is a selection of those forms φf ∈ D(X, Y ) which satisfy: for each x ∈ X there is an open O ∈ B(x) of x such that the closure of the set neighborhood φf (x) : x ∈ O is compact in Y . φf [O] = The spaces of densely continuous forms have probably emerged for the first time in the paper of H a m m e r and M c C o y [HM]. Nowadays, densely continuous forms can be tracked down in various forms in different parts of mathematics, for example as the Argmin multifunction in variational calculus or others, see H o l ´a [Ho1]. They also have an important relationship with minimal USCO maps (see D r e w n o w s k i and L a b u d a [DL]). For example, it has been shown in [HV2] (and earlier in [Ho1] for the real-valued case) that if X is Baire and Y is locally compact metric, then D∗ (X, Y ) is the space of all minimal USCO maps from X to Y . It is well known that the minimal USCO maps are a very convenient tool in the theory of games (see C h r i s t e n s e n [Ch] or S a i n t -R a y m o n d [Sa]) or in functional analysis (see the paper of B o r w e i n and M o o r s [BM] where a differentiability property of single-valued functions is characterized by their Clarke subdifferentials being convex minimal USCO maps). Other properties of DC forms might also be of interest, such as the cardinal functions of topologies on such spaces (see H o l ´a and M c C o y [HoM] or [HV1]) or the selection properties of DC forms (see H o l ´a and H o l y ´ [HH]) or relations of various topologies on spaces of DC forms (see [HV2] or H o l y ´ [Ho]). Moreover, there are other features of minimal USCO maps or densely continuous forms that are worth investigating for more general classes of maps. In particular, the concept of minimality of multifunctions has been successfully generalized and has found interesting applications (see for example the papers of M a t e j d e s [Ma1] and [Ma2]). The concept of generalized (semi)continuity for multifunctions has also been intensively studied (see for example H o l y ´ and M a t e j´ı ˇc k a [HoMa]). 510 Unauthenticated Download Date | 11/24/15 7:50 PM
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2. Properties of DC forms To define the topology τp of pointwise convergence on F (X, 2Y ) let (Y, d) be a metric space and consider the well-known Hausdorff (extended-valued) metric Hd induced by d on the hyperspace 2Y of all closed subsets of Y . The topology τp then has a local base at any φ ∈ F (X, 2Y ) consisting of sets of the form W (φ, A, ε) = ψ ∈ F (X, 2Y ) : (∀ x ∈ A) Hd (φ(x), ψ(x)) < ε for all finite subsets A ∈ F (X) and ε > 0. The subspace D∗ (X, Y ) with the induced topology τp will be denoted by Dp∗ (X, Y ). If Y = R is the space of reals with the usual topology, then the range space is omitted from the notation, i.e. we have D∗ (X) and Dp∗ (X), respectively. Now, if (K (R), H) denotes the space of all nonempty compact subsets of X, K (R) can be seen the reals equipped with the Hausdorff metric, then F p as a subspace of Fp X, 2R . Moreover, we know that every form in D∗ (X) ∗ assumes only nonempty compact values (see [HV2, Lemma 2.2]), so D (X) ⊆ Fp X, K (R) . Then it is not very hard to deduce the following:
2.1
If the Hausdorff space X is without isolated points, then (a) int D(X) = ∅ in Fp X, 2R . (b) int D∗ (X) = ∅ in Fp X, K (R) . P r o o f. It is sufficient to prove (a), since the proof of (b) uses the same basic idea. Suppose by the contrary that there is φ ∈ int D(X) and A ∈ F(X) and ε > 0 such that W (φ, A, ε) ⊆ D(X). Then X \ A is nonempty open. By [Ho1, Proposition 2.2] of H o l a´ , there is an open dense set U ⊆ X such that φ is USCO at every point of U (i.e. φ is upper semicontinuous and has nonempty compact values at the points of U ; see B e e r [Be] or [HV1], [HV2] for definitions). Choose x ∈ U ∩ (X \ A). The set φ(x) is compact, so choose y ∈ / φ(x) and define φ(z), z = x, ψ(z) = φ(x) ∪ {y}, z = x. Hence ψ ∈ Fp X, 2R . There are U , V disjoint open sets such that φ(x) ⊆ U and y ∈ V . Since φ is upper semicontinuous at x, there is an open neighborhood W of x such that φ [ W ] ⊆ U . Since y ∈ / U , this contradicts the idea of ψ belonging to D(X), i.e. of having a generating function, for such function would have to be similar to the generating function of φ. But because of ψ [ W \ {x} ] = φ [ W \ {x} ] ⊆ U , no value outside the set U can be reached at x by such densely continuous form. Therefore ψ ∈ / D(X) but ψ ∈ W (φ, A, ε), a contradiction.
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2.2
If the Hausdorff space X is without isolated points, then (a) D(X) is not closed in Fp X, 2R . (b) D∗ (X) is not closed in Fp X, K (R) . P r o o f. In this case, the proofs of (a) and (b) are actually the same. Choose a point x ∈ X arbitrarily. Define a net of functions in the following way: for each basic open neighborhood U ∈ B(x) let fU be the characteristic function of U in X. For each U ∈ B(x) the boundary of U is nowhere dense in X, so each ∗ fU is a densely continuous function and its induced form φU belongs to D (X). Consider now the (multi)function φ ∈ Fp X, K (R) defined by {0}, y = x, φ(y) = {1}, y = x. Such φ clearly has no generating function, so φ ∈ / D(X). Now let A ∈ F(X) and ε > 0 be arbitrary. Let A∗ = A \ {x}. The set X \ A∗ is an open neighborhood of x, i.e. there is U ∈ B(x) disjoint from A∗ . Then for each V ∈ B(x), V ⊆ U , we have φV A∗ = {0} = φ A∗ and φV (x) = {1} = φ(x), which implies φV ∈ W (φ, A, ε). Therefore the net φU : U ∈ B(x) , which is contained in D∗ (X), converges pointwise to φ ∈ / D(X) and both statements are proved. Example 2.1. The essential difference between locally bounded and ordinary densely continuous forms is that in the general case we are not able to guarantee that all values of the generated densely continuous form are either nonempty (consider the value at 0 of the form generated by the function 1/x in the space R) or are “nice” bounded or compact sets, as the following situation depicts: consider the generating function f (x) = (1/x) · sin (1/x) for x = 0 which is a continuous single-valued function everywhere except for one point of the domain, and its generated form is 1 1 , x = 0, x · sin x φ(x) = R, x = 0. This is the major obstacle (sometimes invincible) in finding analogs of the results of locally bounded densely continuous forms for ordinary densely continuous forms.
3. Cellularity of Dp∗ (X): Fr´ echet and Volterra spaces In this section we investigate a property of first countable regular spaces which allows the construction of the desired densely continuous form in the proof of the density lemma ([HV1, Lemma 3.12]), saying that for a first countable regular 512 Unauthenticated Download Date | 11/24/15 7:50 PM
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space (without isolated points) Dp∗ (X) is dense in Fp X, K (R) . The corollary of this lemma ([HV1, 3.13]) then implies the countable cellularity of Dp∗ (X) for first countable regular spaces X. Subsequently we shall discuss some applications of the investigated property for densely continuous forms on Fr´echet and Volterra spaces as well as some improvements of the mentioned sufficient condition from [HV1] concerning countable cellularity of Dp∗ (X).
3.1
We say the topological space X has the property P, if:
∀ x ∈ X ∀ Ox ∈ B(x) ∀ l ∈ N ∃ {Ui }∞ i=1 pairwise disjoint open sets such that Ui ⊆ Ox and x ∈ Ui for each i = 1, . . . , l. Recall that a topological space is a Fr´echet space (see [En]) if for any point x and any subset with x ∈ A, there is a sequence in A converging to x.
3.1
Let X be a Hausdorff and Fr´echet space without isolated points. Then X has the property P.
P r o o f. Let l ∈ N, let x ∈ X and, without the loss of generality, let Ox ∈ B(x) be its non-trivial neighborhood. Let {xn : n ∈ N} be a sequence of distinct points of Ox converging to x. Since the set Ck = {x} ∪ {xn : n > k} is compact and closed for each k ∈ N we are able to define by induction the open disjoint sets Ok and Vk such that Ck ⊆ Vk ⊆ Vk−1 and xk ∈ Ok ⊆ Vk−1 (of course put V0 = Ox ). Thus we have a sequence {On : n ∈ N} of pairwise disjoint open sets with xn ∈ On . Now, for each i = 1, . . . , l put {On : n ∈ N, n mod l = i − 1}. Ui = Then x ∈ Ui , so {Ui }li=1 is the desired pairwise disjoint system of open sets.
Note 3.1 A topological space X is a Volterra space (see [GP]), if for each f, g ∈ DC(X, R) the set C(f ) ∩ C(g) is dense in X. It was shown in [GGP1] that X is Volterra if and only if for each pair A, B ⊆ X of dense Gδ subsets the set A ∩ B is dense. Thus, every Baire space is Volterra, but there are Volterra spaces which are not of second category, i.e. not Baire spaces (see [GGP2]). It can be shown (see H o l ´a [Ho2] or M c C o y [McC]) that if X is a Volterra (or Baire) space, then we can make the space D(X) of densely continuous forms a vector space by defining: for densely continuous forms φf = f C (f ), φg = g C (g) put φf + φg := f + g C (f + g) = φf +g and for each a ∈ R put a · φf := a · f C (a · f ) = φa·f . 513 Unauthenticated Download Date | 11/24/15 7:50 PM
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3.1
Let X be a Hausdorff, Fr´echet and Volterra space with a nonisolated point. Then the addition is not continuous on Dp∗ (X), that is, Dp∗ (X) is not a linear topological space. P r o o f. Let x ∈ X be the non-isolated point, let Ox ∈ B(x) be its non-trivial open neighborhood, let l = 2. Then X has the property P at the point x ∈ Ox , i.e. using the procedure from Lemma 3.1 we get the disjoint open subsets {U1 , U2 } in Ox such that x ∈ U1 ∩ U2 . Define the functions f, g : X → R by 1, y ∈ U1 , f (y) = −1, y ∈ / U1 , −1, y ∈ U1 , g(y) = 1, y ∈ / U1 . Then f, g ∈ DC(X) and we have φf = f C (f ), φg = g C (g) in Dp∗ (X) such that φf +g ≡ 0 ≡ φ0 is the zero (multi) function in the vector space with addition defined as in the remarks preceding the theorem. Now if A ∈ F(x) is any finite subset of X and ε > 0, we want to find an / W (φ0 , {x}, 1). So element φh ∈ Dp∗ (X) such that φh ∈ W (φf , A, ε) but φh+g ∈ let A = A \ {x}. Since A is compact, there is an open neighborhood U ∈ B(x) of x such that U ∩ A = ∅. Define the function h : X → R by g(y), y ∈ U, h(y) = f (y), y ∈ / U. Then again h ∈ DC(X) and we have φh ∈ Dp∗ (X). If y ∈ A , then y ∈ / U, so φh (y) = φf (y). Moreover, since x ∈ U ∈ B(x), it is φh (x) = φg (x) = {−1, 1} = φf (x) and therefore φh ∈ W (φf , A, ε). On the other hand, φh +φg = φh+g and (h+g)(y) = 2g(y) for y ∈ U and therefore (φh + φg )(x) = 2φg (x) = {−2, 2}, which implies φh + φg ∈ / W (φ0 , {x}, 1). This completes the proof of the theorem. On the other hand, one is able to see that the scalar multiplication is continuous on Dp∗ (X) even for non-Fr´echet spaces. This is due to the fact that the finitely-determined basic sets W (φf , A, ε) in Dp∗ (X) are too “loose” in determining the properties of a contained form φg , but are still sufficient when the form is multiplied by a real scalar. 514 Unauthenticated Download Date | 11/24/15 7:50 PM
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3.2
Let X be a Hausdorff and Volterra space. Then the scalar multiplication is continuous on Dp∗ (X). P r o o f. Let φf ∈ Dp∗ (X), A ∈ F(X) be finite and let ε > 0. Let a ∈ R be a scalar (without the loss of generality we may suppose that a = 0. It suffices to find δ > 0 such that for each b ∈ Sδ (a) := (a − δ, a +δ) and each φg ∈ Sδ (a) : a ∈ A . W (φf , A, δ) we have φb·g ∈ W (φa·f , A, ε). Denote Sδ [A] := Since φf assumes only nonempty compact values (see the remarks preceding Theorem 2.1), the set φf [A] is compact, so we can find M = max max |y| + ε : y ∈ φf [A] , |a| + 1 . ε Moreover M ≥ max {1, ε} and if we put δ = 2M , then δ ≤ min ε2 , 12 . Now, let φg ∈ W (φf , A, δ), let b ∈ Sδ (a) and let x ∈ A. We want to show that H (φb·g (x) , φa·f (x)) < ε and for that it is sufficient to find for each s ∈ φa·f (x) an element t ∈ φb·g (x) such that |s − t| < ε and vice versa (this argument will be symmetrical and will be dealt with later). So if s ∈ φa·f (x), then (x, s) ∈ φa·f = a · f C (f ) and for each U ∈ B(x) and each V ∈ B(s) we have x(U,V ) ∈ U ∩ C (f ) such that a · f (x(U,V ) ) ∈ V . Thus we have a pair of double-indexed nets: {x(U,V ) } converging to x and {a · f (x(U,V ) )} converging to s. Put s = as . Since Y = R is a topological group, if V ∈ B(s ), then a · V ∈ B(s), i.e. the net {f (x(U,V ) )} converges to s and therefore (x, s ) ∈ f C (f ) = φf or equivalently s ∈ φf (x). Recalling that H (φg (x) , φf (x)) < δ, we have φf (x) ⊆ Sδ [φg (x)]
and
φg (x) ⊆ Sδ [φf (x)] .
Thus there is t ∈ φg (x) such that |s − t | < δ. Since (x, t ) ∈ φg = g C (g), the same argument as before implies that (x, b · t ) ∈ φb·g = b · g C (g), i.e t := b · t ∈ φb·g (x). Finally, since t ∈ Sδ [φf (x)], there is y ∈ φf (x) such that |t | ≤ |y| + δ < |y| + ε ≤ M . Now we have |s − t| = |a · s − b · t | ≤ |a| · |s − t | + |t | · |a − b| ε < M · δ + M · δ = 2M · = ε. 2M Considering the reverse implication, we apply the same approach and for t ∈ φb·g (x) we get s ∈ φa·f (x) with |t − s| = |b · t − a · s | ≤ |b| · |t − s | + |s | · |b − a| < ε, because |b| < |a| + δ < |a| + 1 ≤ M . Therefore H (φb·g (x) , φa·f (x)) < ε and the theorem is proved. 515 Unauthenticated Download Date | 11/24/15 7:50 PM
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The fact that a Fr´echet space possesses the property P is also important for our next goal, the generalization of the countable cellularity of Dp∗ (X). First, realize the following theorem:
3.3 If X is a Hausdorff space with the property P and without isolated points, then Dp∗ (X) is dense in Fp X, K (R) . P r o o f. Let W (ψ, A, ε) be an arbitrary nonempty open basic set in Fp X, K (R) where A = {x1 , . . . , xm } is finite and ε > 0. Each ψ(xi ) is compact, so for each i = 1, . . . , m there is a finite subset Yi = {y1 , . . . , yli } in ψ(xi ) such that ψ(xi ) ⊆ Sε/2 [Yi ]. Since A is finite, for each i = 1, . . . , m there is a non-trivial open neighborhood O(xi ) of xi such that O(xi ) ∩ O(xj ) = ∅, if i = j. Since X has the property P, xi ∈ O(xi ) and li ∈ N, for each i ∈ {1, . . . , m} we have the family {U1 , . . . , Uli } of pairwise disjoint open sets in O(xi ) such that xi ∈ Ui . On each O(xi ) define the function f : X → R by yk , y ∈ Uk , k = 1, . . . , li , f (y) = z, y ∈ / {Uk : k = 1, . . . , li }, where z ∈ Yi = {y1 , . . . , yli } is arbitrary fixed. Outside the set
m
O(xi ) the
i=1
function f can be put arbitrarily constant. Then f ∈ DC(X) is densely continuous and let φ = f C (f ) ∈ Dp∗ (X). The definition of f implies φ(xi ) = {y1 , . . . , yli } = Yi ⊆ ψ(xi ). By our initial remarks: ψ(xi ) ⊆ Sε/2 [φ (xi )] and therefore H(φ(xi ), ψ(xi )) ≤ ε/2 < ε for every i = 1, . . . , m, or equivalently, φ ∈ Dp∗ (X) ∩ W (ψ, A, ε) which proves the theorem. Now we are ready to deliver the main result — a generalization of the sufficient condition for the countable cellularity of Dp∗ (X) (see [HV1, Theorem 3.13]). For the definition of cellularity see the next section of the paper. 3.4 Let X be a Hausdorff space which has the property P. Then c Dp∗ (X) = ℵ0 . P r o o f. If X contains isolated points, then the construction described in the preceding theorem cannot be applied directly, because at an isolated point any densely continuous form is single-valued, i.e. no larger compact set can be generated as a value at such a point by a densely continuous form. 516 Unauthenticated Download Date | 11/24/15 7:50 PM
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Thus consider the following: let I(X) denote the set of all isolated points in
X and denote by Z = Yx the product space where Yx = R (with the usual x∈X
topology) for x ∈ I(X) and Yx = (K (R), H) if x ∈ / I(X). We know by a result of E n g e l k i n g (see [En, Corollary 2.3.18]) that the cellularity of Z is countable, and we know that cellularity is hereditary for dense subspaces. So let W (ψ, A, ε) be an arbitrary nonempty open set in Z, i.e. ψ ∈ Z is a form such that ψ is single-valued on I(X) and compact-valued on X \ I(X). Define the function f : X → R by f (x) = ψ(x) for x ∈ I(X) and define f on X \ I(X) as in the proof of the previous theorem. Then φ = f C (f ) is a locally bounded densely continuous form which is ε-close to ψ on A, i.e. Dp∗ (X) is dense in Z and hence c Dp∗ (X) = c (Z) = ℵ0 . ∗ 3.1 If X is a Hausdorff and Fr´echet space, then c Dp (X) = ℵ0 . Note 3.2 It can be observed that if X is the product of any number of Hausdorff Fr´echet spaces without isolated points, then X also has the property P.
3.2 If X is the product of any number of Hausdorff Fr´echet spaces, then c Dp∗ (X) = ℵ0 .
4. Cardinal functions: Dp∗ (X) vs. Cp (X) To close this paper of remarks we intend to perform a short comparison of cardinal functions on two spaces equipped with the topology of pointwise convergence: the space Dp∗ (X) of real-valued locally bounded densely continuous forms on X and the space Cp (X) of continuous real-valued functions on X. For the first part, the most common cardinal functions for Dp∗ (X) were determined (or have had boundaries found) in the paper [HV1]. Concerning Cp (X), we will use the standard arguments and the knowledge of Cp -theory (see for example the books of A r h a n g e l s k i i [Ar] or M c C o y and N t a n t u [MN]). Moreover, all cardinal functions are supposed to assume only infinite values (as in E n g e l k i n g [En]) and to conform with the assumptions of Cp -theory, the space X is throughout the section supposed to be Tychonoff (completely regular T1 ). We shall consider classic cardinal functions like the weight w(Z), the network weight nw(Z) or the density d(Z) of a space Z which are defined as the minimal cardinalities of all bases, networks or dense subsets, respectively. The cellularity c(Z), on the other hand, is defined as the supremum of the cardinalities of all systems of pairwise disjoint open sets. Firstly, it follows from [HV1, Theorem 3.8] and from [MN, Theorem 4.5.2] of M c C o y and N t a n t u that the weight satisfies: 517 Unauthenticated Download Date | 11/24/15 7:50 PM
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4.1 For every Tychonoff space X, w
Dp∗ (X) = |X| = w (Cp (X)).
Next, if we consider the pseudocharacter of the space Z defined by Ψ(Z) = sup Ψ(Z, z) : z ∈ Z , where Ψ(Z, z) is the minimum of the cardinalities of all families G of open sets in Z such that G = {z}, and if we consider the diagonal degree ∆(Z) of Z defined as the minimum of all families G of the cardinalities of open sets in Z × Z such that G = (z, z) : z ∈ Z , then it follows from [HV1, Theorem 3.2] and from [MN, Theorem 4.3.1] that:
4.2
For every Tychonoff space X, ∗ Ψ Dp (X) = Ψ (Cp (X)) = ∆ Dp∗ (X) = ∆ (Cp (X)) = d (X) .
Continuing along this line, recall that a family P(z) of nonempty open sets is called a local π-base (local pseudo-base) at z ∈ Z, if for each U ∈ B(z) there is P ∈ P(z) with P ⊆ U . Then the π-character of Z is defined by πχ (Z) = sup πχ (Z, z) : z ∈ Z , where πχ (Z, z) is the minimum of the cardinalities of all local π-bases at z ∈ Z. The character of Z is defined by χ(Z) = sup χ(Z, z) : z ∈ Z , where χ(Z, z) is the minimum of the cardinalities of all local bases at z ∈ Z. It then follows from [HV1, Theorem 3.5] and from [MN, Theorem 4.4.1] that:
4.3 πχ
For every Tychonoff space X, Dp∗ (X) = πχ (Cp (X)) = χ Dp∗ (X) = χ (Cp (X)) = |X|.
Concerning the cellularity, from the results of the previous section and from the results of Cp -theory we have the following corollary.
4.4 If X is the product of Fr´echet Tychonoff spaces, then we have c Dp∗ (X) = ℵ0 = c (Cp (X)). These results might lead to the conclusion that all the cardinals on Dp∗ (X) and Cp (X) are in fact the same. Such conclusion, however, is quickly proved wrong by the virtue of [HV1, Example 4.1], which states that if X = R is the space of all reals with the usual topology, then d Dp∗ (R) > ℵ0 . It follows that:
4.5 Whenever I is a compact interval of the reals, then the density
satisfies:
d Dp∗ (I) > ℵ0 = d (Cp (I)) .
And directly from [HV1, Example 4.1] and [MN, Theorem 4.1.2] it also follows that:
4.6
The network weight satisfies nw Dp∗ (R) > ℵ0 = nw (Cp (R)) .
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Note 4.1 The collection of all results of [HV1] together with [HV2] and some previous results concerning function-space topologies can be found in [Va]. a for suggesAcknowledgement The author would like to thank L’ubica Hol´ tions and comments and also Georges Grekos of LaMUSE, France.
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ˇ PETER VADOVIC [Ma1] [Ma2] [McC] [MN] [Sa] [Va]
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Received 29. 12. 2006 Revised 6. 9. 2007
Mathematical Institute Slovak Academy of Sciences ˇ anikova 49 Stef´ SK–81473 Bratislava SLOVAKIA
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