Journal of Applied Analysis Vol. 5, No. 1 (1999), pp. 71–94
COVERING BAIRE 1 FUNCTIONS WITH DARBOUX FUNCTIONS AND THE COFINALITY OF THE IDEAL OF MEAGER SETS F. JORDAN Received February 2, 1998 and, in revised form, September 23, 1998
Abstract. Let Dar stand for the Darboux Baire class 1 functions. We show that the cofinality of the meager sets in R is the smallest cardinality of a set of Baire class 1 functions F such that for any finite collection of Baire class 1 functions G there is an f ∈ F such that f + G ⊆ Dar. Other results of this type are shown. These results are then considered as statements about additivity. The notion of super–additivity is introduced.
1. Preliminaries We use standard notation as in [5]. In particular, for a set X we denote its cardinality by |X|. Given sets X and Y we denote by Y X the set of all functions from X into Y . If κ is a cardinal number and X is a set we let [X] 0 and neighborhood W of x0 there is a nonempty open set W0 ⊆ W such that osc(f, W0 ) < ε. We denote the family of cliquish functions by cliq. It is well known, and easy to prove, that if f ∈ cliq then
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cont(f ) is a co-meager subset of R. We note the following facts about Baire class one and cliquish functions. Proposition 2. (1) B1 ⊆ cliq, (2) oscn (f ) is closed and nowhere dense for any f ∈ cliq and n ∈ ω, (3) B1 and cliq are additive groups. Proof. The containment B1 ⊆ cliq follows from [13, p.394]. Item (2) follows immediately from Proposition 1. The cliquish functions form an additive group since, if f, g ∈ cliq then cont(f + g) ⊇ cont(f ) ∩ cont(g) and cont(f ) ∩ cont(g) must be a co-meager set by the Baire Category Theorem. The Baire class one functions form an additive group because the sum of the limits of two pointwise convergent sequences of functions is equal to the pointwise limit of the sums of the terms of the sequences.
2. Introduction We will be concerned with some cardinal invariants of the real line related to the algebraic properties of the following families of functions in B1 and cliq. We give descriptions of these families for general topological spaces, although our discussion will be restricted to the real line. To find out more about the families below see [9], [17], and [6]. Dar: f ∈ Y X is a Darboux function if and only if f [C] is connected in Y for every connected subset C of X. Con: f ∈ Y X is a connectivity function if and only if the graph of f restricted to C is connected in X × Y for every connected subset C of X. Ac: f ∈ Y X is an almost continuous function if and only if every open set in X × Y containing f also contains some continuous function g ∈ Y X . Ext: f ∈ Y X is an extendable function if and only if there is a connectivity function g : X × [0, 1] → Y such that f (x) = g(0, x) for every x ∈ X. Pr: f ∈ RR is a function with a perfect road if and only if for every x ∈ R there is a perfect set P such that x is a bilateral limit point of P and the restriction f |P is continuous at x. Pc: f ∈ Y X is a peripherally continuous function if and only if for every x ∈ X and every pair of open sets U ⊂ X and V ⊂ Y such that x ∈ U and f (x) ∈ V there is an open neighborhood W of x with cl(W ) ⊂ U and f [bd(W )] ⊆ V , where bd(W ) denotes the boundary of W . Qc: f ∈ Y X is a quasi-continuous function if and only if at each point p ∈ X the following condition holds: for every open set U ⊆ X with p ∈ U and open set V ⊆ Y with f (p) ∈ V there exists a non-empty open set W ⊆ U such that f [W ] ⊆ V .
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´ atkowski function if and only if for every a < b Sw: f ∈ RR is a strong Swi¸ and y strictly between f (a) and f (b) there is an x ∈ cont(f ) ∩ (a, b) such that f (x) = y. We collect some facts about the above classes which will be of use in what follows. Proposition 3. The following containments and equalities hold and all containments mentioned are proper. (i) If F ∈ {Ext, Ac, Pr, Con, Pc} then Dar ∩ B1 = F ∩ B1 , (ii) Sw ⊆ Dar ∩ Qc, (iii) Qc ⊆ cliq, (iv) Qc ∩ Pr = Qc ∩ Pc, (v) Pr ∪ Dar ⊆ Pc. Proof. The containment of (ii) is straightforward and left without proof. The following example [15, p.10] shows that the containment of (ii) is proper if x > 0; 1 + x + sin(1/x) g(x) = −1 + x + sin(1/x) if x < 0; 0 if x = 0.
It is clear from the definitions that Qc ⊆ cliq, the characteristic function of a point shows that the containment is proper. Thus, (iii) holds. We give the citations for (i). The equalities Dar ∩ B1 = F ∩ B1 for F = Ext, Ac, Pr, Con, Pc are shown in [3], [2], [16], [14], and [18] respectively. The equality Qc ∩ Pr = Qc ∩ Pc is shown in [8]. For (v) see [3] and [8]. In [15, p.17] A. Maliszewski proved. 0 we may assume that norm(f ) ≤ sup{osc(g, x) : g ∈ H& x ∈ R} + ε.
It was also shown in [15, p.19] that the above proposition could not be improved to include infinite families of functions. Proposition 5. For any function g, if cont(g) 6= ∅ then there is a q ∈ such that g + χ{q} ∈ / Dar ∪ Qc.
Q
The two propositions above may be interpreted as statements about a cardinal function which has been studied for families of real functions in
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more general settings [6]. Given a family F ⊆ RR the additivity of F, denoted by A(F), is defined to be A(F) = min {|F | : F ⊆ RR & ∀g ∈ RR (∃f ∈ F )(f + g ∈ / F )} ∪ {(2c )+ } . The above definition may be restricted to smaller classes of functions in the following way. Let H ⊆ RR and F ⊆ RR then the additivity of F in H, denoted by AH (F), is defined to be AH (F) = min {|F | : F ⊆ H & (∀g ∈ H) (∃f ∈ F )(f + g ∈ / F )} ∪ {|H|+ } .
The following proposition is an adaptation of [11, Proposition 1] which dealt with the special case H = RR . Proposition 6. Let P, F, H ⊆ RR and χ∅ ∈ H. Then, (i) if F ∩ H = ∅, then AH (F) = 1; (ii) if F ∩ H = H, then AH (F) = |H|+ ; (iii) if F ⊆ P, then AH (F) ≤ AH (P); (iv) if H is an additive group with more than one element and F ∩H= 6 ∅, then 2 = AH (F) if and only if (F ∩ H) − (F ∩ H) 6= H; (v) if 2 < AH (F), then (F − F) ∩ H = H.
Proof. We show (i). If F ∩H = ∅, then χ∅ has the property that χ∅ +h ∈ /F for all h ∈ H. Thus, AH (F) = 1. We show (ii). Suppose AH (F) < |H|+ . Then there is an F ⊆ H with the property that for every h ∈ H there is some f ∈ F such that f + h ∈ / F. In particular, there is some f ∈ F ⊆ H such that f = f + χ∅ ∈ / F. Since the proof of (iii) differs only slightly from the proof of its equivalent statement in [11, Proposition 1], we exclude the proof. We show (iv). Suppose that (F ∩ H) − (F ∩ H) = H. We consider the two possible cases. Let S = {F ⊆ H : (∀g ∈ H)(∃f ∈ F )(f + g ∈ / F)}. First assume S = ∅. Then AH (F) = |H|+ > 2. Now assume that there exists an F ∈ S. We show that |F | > 2. By way of contradiction, assume that F = {f1 , f2 } ∈ [H] 2. To see the other implication suppose that (F ∩ H) − (F ∩ H) 6= H. Since H is a group and F ∩ H 6= ∅ it is easy to see that AH (F) ≥ 2. So we show that AH (F) ≤ 2. Pick h ∈ H \ ((F ∩ H) − (F ∩ H)) and put F = {h, χ∅ }. Let g ∈ H be arbitrary. It is enough to show that f + g ∈ / F for some f ∈ F . However, if g = χ∅ + g ∈ F and h + g ∈ F then, since H is a group, we have h ∈ (F ∩ H) − g ⊆ (F ∩ H) − (F ∩ H), contradicting the choice of h.
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We prove (v). Suppose that (F − F) ∩ H 6= H. Pick h ∈ H \ (F − F) and let F = {χ∅ , h}. By way of contradiction, assume there is some g ∈ H such that g + F ⊆ F. Then, g = g + χ∅ ∈ F , so h ∈ F − g ⊆ (F − F) which contradicts the choice of h. Thus, AH (F) ≤ 2. Additivity and its restricted versions have a nice interpretation in terms of coverings of one family of functions by another. We state this relationship in the following proposition: Proposition 7. Let F, H ⊆ RR and AH (F) < |H|T+ . Then AH (F) is the minimum cardinality of a set that H ∩ {−f + F : f ∈ F } = ∅; S F ⊆ H such R or, equivalently, that H ⊆ {−f + (R \ F ) : f ∈ F }. Proof. Since AH (F) < |H|+ , there is an F ⊆ H that witnesses the main part of the definition of AH (F), i.e., |F | = AH (F) and (∀h ∈ H)(∃f ∈ F )(h + f ∈ / F ). (1) T We claim thatTH ∩ {−f + F : f ∈ F } = ∅. To see this, assume there is some h ∈ H ∩ {−f + F : f ∈ F }. Such an h would have the property that h + f ∈ F for each f ∈ F . But this would contradict (1) since h ∈ H. Thus, \ H ∩ {−f + F : f ∈ F } = ∅
for some F of cardinality AH (F). Now assume F ⊆ H and |F | < AH (F). There is T an h ∈ H such that h+F ⊆ F. So, h ∈ −f +F for each f ∈ F . Thus, H∩ {−f +F : g ∈ F } 6= ∅ for any F such that |F | < AH (F), which completes the proof.
Using the language of additivity, we will now state some corollaries to Propositions 4 and 5. Corollary 8. If F ∈ {Sw, Dar ∩ Qc, Dar, Qc, Dar ∪ Qc}, then Acliq (F) = AB1 (F) = ω. Proof. By containment, we have that Acliq (Sw) ≤ Acliq (F) ≤ Acliq (Qc ∪ Dar) for F ∈ {Dar, Qc, Sw, Dar ∩ Qc, Dar ∪ Qc}. Since B1 ⊆ cliq, it follows from Proposition 4 that ω ≤ Acliq (Sw). Using Proposition 5, the fact that any cliquish function must have at least one point of continuity, and noting that the characteristic function of a point is cliquish, we have Acliq (Dar ∪ Qc) ≤ ω. It now follows that ω = Acliq (F) for all F ∈ {Dar, Qc, Sw, Dar ∩ Qc, Dar ∪ Qc}. A similar argument yields the equalities for AB1 .
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Corollary 8 together with Proposition 7 implies that the minimum number of translations of B1 \ Dar by Baire class one functions required to cover Baire class one is ω. To get a statement similar to that of Corollary 8 for quasi-continuous functions, we need to make a minor modification of Proposition 5. In particular, the characteristic functions of singletons are not quasi-continuous so Proposition 5 will not be useful to us in the quasi-continuous case. So, we will consider characteristic functions of open intervals, which are quasicontinuous. It should be pointed out that the proof here is essentially identical to the one which appears in [15] for Proposition 5. Proposition 9. For any function g, if cont(g) 6= ∅, then there is a q ∈ and an n ∈ ω such that g + χ(q−1/n,q+1/n) ∈ / Dar.
Q
Proof. Suppose g is a function such that cont(g) 6= ∅. Let x ∈ cont(g). There is a δ > 0 such that g[(x − δ, x + δ)] ⊆ (g(x) − 1/3, g(x) + 1/3). Pick q ∈ Q and n ∈ ω such that [q − 1/n, q + 1/n] ⊆ (x − δ, x + δ). For w ∈ (q − 1/n, q + 1/n), we have (g + χ(q−1/n,q+1/n) )(w) > 1 + g(x) − 1/3 = g(x) + 2/3.
For points z ∈ (x − δ, x + δ) \ (q − 1/n, q + 1/n), we have (g + χ(q−1/n,q+1/n) )(z) < g(x) + 1/3.
Thus, (g + χ(q−1/n,q+1/n) )[(x − δ, x + δ)] is not an interval. Therefore, we must conclude that g + χ(q−1/n,q+1/n) is not Darboux. Corollary 10. AB1 ∩Qc (Sw) = AB1 ∩Qc (Dar) = AQc (Dar) = AQc (Sw) = ω. Proof. We first show that AQc (Sw) = AQc (Dar) = ω. Since Sw ⊆ Qc ⊆ cliq, it follows from Proposition 4 that ω ≤ AQc (Sw). The countable family of functions constructed in Proposition 9 were all quasi-continuous and any quasi-continuous function has a point of continuity; so, we have AQc (Dar) ≤ ω. Finally, we have AQc (Sw) ≤ AQc (Dar). Thus, AQc (Sw) = AQc (Dar) = ω. A similar argument will give the equalities for AQc∩B1 . Corollary 8 tells us that for any finite collection G of Baire one functions there is an f ∈ B1 such that f + G ⊆ Dar. One may ask what the minimal cardinality of a family of Baire one functions F such that for every finite family of functions G there is an f ∈ F such that f + G ⊆ Dar is. This question leads us to define a new cardinal function which, like additivity, has meaning in more general settings [12] and also has a nice interpretation
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in terms of coverings. If H, F ⊆ RR , then the super-additivity of F in H is defined to be n o A∗H (F) = min |F | : F ⊆ H & ∀G ∈ [H] 0. Thus, (25) is established, and so f + g ∈ Qc ∩ Dar.
Proof of Theorem 15. Notice that in the proof of Theorem 14 the only place we needed to use the fact that f + H ⊆ B1 was when we used (a) of Lemma 26 to say that (25) implied f + g ⊆ Dar ∩ Qc. So, if we had just wanted to prove that f + H ⊆ Qc ∩ Pr, we could have just assumed that H ⊆ cliq and used (b) of Lemma 26 to say that f + g ∈ Qc ∩ Pr.
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Francis Jordan Department of Mathematics University of Louisville Louisville, KY 40208 USA email address:
[email protected] 1 Preprints marked by * are available in electronic form. They can be accessed from Set Theoretic Analysis Web Page: http://www.math.wvu.edu/homepages/kcies/STA/STA.html.
