() @ Applied General Topology c© Universidad Politécnica de Valencia Volume 13, no. 2, 2012 pp. 167-178 Classification of separately continuous mappings with values in σ-metrizable spaces Olena Karlova Abstract We prove that every vertically nearly separately continuous mapping defined on a product of a strong PP-space and a topological space and with values in a strongly σ-metrizable space with a special stratification, is a pointwise limit of continuous mappings. 2010 MSC: 54C08, 54C05, 54E20 Keywords: separately continuous mapping, σ-metrizable space, strong PP- space, Baire classification, Lebesgue classification 1. Introduction Let X, Y and Z be topological spaces. By C(X, Y ) we denote the collection of all continuous mappings from X to Y . For a mapping f : X × Y → Z and a point (x, y) ∈ X × Y we write f x(y) = fy(x) = f(x, y). We say that a mapping f : X × Y → Z is separately continuous, f ∈ CC(X×Y, Z), if fx ∈ C(Y, Z) and fy ∈ C(X, Z) for every point (x, y) ∈ X×Y . A mapping f : X ×Y → Z is said to be vertically nearly separately continuous, f ∈ CC(X × Y, Z), if fy ∈ C(X, Z) for every y ∈ Y and there exists a dense set D ⊆ X such that fx ∈ C(Y, Z) for all x ∈ D. Let B0(X, Y ) = C(X, Y ). Assume that the classes Bξ(X, Y ) are already defined for all ξ < α, where α < ω1. Then f : X → Y is said to be of the α-th Baire class, f ∈ Bα(X, Y ), if f is a pointwise limit of a sequence of mappings fn ∈ Bξn(X, Y ), where ξn < α. In particular, f ∈ B1(X, Y ) if it is a pointwise limit of a sequence of continuous mappings. 168 O. Karlova In 1898 H. Lebesgue [12] proved that every real-valued separately continuous function of two real variables is of the first Baire class. Lebesgue’s theorem was generalized by many mathematicians (see [4, 15, 17, 19, 18, 1, 2, 5, 6, 16] and the references given there). W. Rudin[17] showed that CC(X × Y, Z) ⊆ B1(X × Y, Z) if X is a metrizable space, Y a topological space and Z a locally convex topological vector space. Naturally the following question has been arose, which is still unanswered. Problem 1.1. Let X be a metrizable space, Y a topological space and Z a topological vector space. Does every separately continuous mapping f : X×Y → Z belong to the first Baire class? V. Maslyuchenko and A. Kalancha [5] showed that the answer is positive, when X is a metrizable space with finite Čech-Lebesgue dimension. T. Banakh [1] gave a positive answer in the case that X is a metrically quarter-stratifiable paracompact strongly countably dimensional space and Z is an equiconnected space. In [8] it was shown that the answer to Problem 1.1 is positive for metrizable spaces X and Y and a metrizable arcwise connected and locally arcwise connected space Z. It was pointed out in [9] that CC(X × Y, Z) ⊆ B1(X × Y, Z) if X is a metrizable space, Y is a topological space and Z is an equiconnected strongly σ-metrizable space with a stratification (Zn) ∞ n=1 (see the definitions below), where Zn is a metrizable arcwise connected and locally arcwise connected space for every n ∈ N. In this paper we generalize the above-mentioned result from [9] to the case of vertically nearly separately continuous mappings. To do this, we intro- duce the class of strong PP-spaces which includes the class of all metrizable spaces. In Section 3 we investigate some properties of strong PP-spaces. In Sec- tion 4 we establish an auxiliary result which generalizes the famous Kuratowski- Montgomery theorem (see [11] and [14]). Finally, in Section 5 we prove that the inclusion CC(X ×Y, Z) ⊆ B1(X ×Y, Z) holds if X is a strongly PP-space, Y is a topological space and Z is a contractible space with a stratification (Zn) ∞ n=1, where Zn is a metrizable arcwise connected and locally arcwise connected space for every n ∈ N. 2. Preliminary observations A subset A of a topological space X is a zero (co-zero) set if A = f−1(0) (A = f−1((0, 1])) for some continuous function f : X → [0, 1]. Let G∗0 and F ∗ 0 be collections of all co-zero and zero subsets of X, respectively. Assume that the classes G∗ξ and F ∗ ξ are defined for all ξ < α, where 0 < α < ω1. Then, if α is odd, the class G∗α (F ∗ α) is consists of all countable intersections (unions) of sets of lower classes, and, if α is even, the class G∗α (F ∗ α) is consists of all countable unions (intersections) of sets of lower classes. The classes F∗α for odd α and G∗α for even α are said to be functionally additive, and the classes F∗α for even α and G ∗ α for odd α are called functionally multiplicative. If a set belongs to the α’th functionally additive and functionally multiplicative class, Classification of separately continuous mappings 169 then it is called functionally ambiguous of the α’th class. Note that A ∈ F∗α if and only if X \ A ∈ G∗α. If a set A is of the first functionally additive (multiplicative) class, we say that A is an F ∗σ (G ∗ δ) set. Let us observe that if X is a perfectly normal space (i.e. a normal space in which every closed subset is Gδ), then functionally additive and functionally multiplicative classes coincide with ordinary additive and multiplicative classes respectively, since every open set in X is functionally open. Lemma 2.1. Let α ≥ 0, X be a topological space and let A ⊆ X be of the α’th functionally multiplicative class. Then there exists a function f ∈ Bα(X, [0, 1]) such that A = f−1(0). Proof. The hypothesis of the lemma is obvious if α = 0. Suppose the assertion of the lemma is true for all ξ < α and let A be a set of the α’th functionally multiplicative class. Then A = ∞ ⋂ n=1 An, where An belong to the αn’th functionally additive class with αn < α for all n ∈ N. By assumption, there exists a sequence of functions fn ∈ Bαn(X, [0, 1]) such that An = f −1 n ((0, 1]). Notice that for every n the characteristic function χAn of An belongs to the α-th Baire class. Indeed, setting hn,m(x) = m √ fn(x), we obtain a sequence of functions hn,m ∈ Bαn(X, [0, 1]) which is pointwise convergent to χAn. Now let f(x) = 1 − ∞ ∑ n=1 1 2n χAn(x). for all x ∈ X. Then f ∈ Bα(X, [0, 1]) as a sum of a uniform convergent series of functions of the α’th class. Moreover, it is easy to see that A = f−1(0). � A topological space X is called • equiconnected if there exists a continuous function λ : X × X × [0, 1] → X such that (1) λ(x, y, 0) = x; (2) λ(x, y, 1) = y; (3) λ(x, x, t) = x for all x, y ∈ X and t ∈ [0, 1]. • contractible if there exist x∗ ∈ X and a continuous mapping γ : X × [0, 1] → X such that γ(x, 0) = x and γ(x, 1) = x∗. A contractible space X with such a point x∗ and such a mapping γ is denoted by (X, x∗, γ). Remark that every convex subset X of a topological vector space is equicon- nected, where λ : X × X × [0, 1] → X is defined by the formula λ(x, y, t) = (1 − t)x + ty, x, y ∈ X, t ∈ [0, 1]. It is easily seen that a topological space X is contractible if and only if there exists a continuous mapping λ : X × X × [0, 1] → X such that λ(x, y, 0) = x and λ(x, y, 1) = y for all x, y ∈ X. Indeed, if (X, x∗, γ) is a contractible space, 170 O. Karlova then the formula λ(x, y, t) = { γ(x, 2t), 0 ≤ t ≤ 1 2 , γ(y, −2t + 2), 1 2 < t ≤ 1. defines a continuous mapping λ : X ×X ×[0, 1] → X with the required proper- ties. Conversely, if X is equiconnected, then fixing a point x∗ ∈ X and setting γ(x, t) = λ(x, x∗, t), we obtain that the space (X, x∗, γ) is contractible. Lemma 2.2. Let 0 ≤ α < ω1, X a topological space, Y a contractible space, A1, . . . , An be disjoint sets of the α’th functionally multiplicative class in X and fi ∈ Bα(X, Y ) for each 1 ≤ i ≤ n. Then there exists a mapping f ∈ Bα(X, Y ) such that f|Ai = fi for each 1 ≤ i ≤ n. Proof. Let n = 2. In view of Lemma 2.1 there exist functions hi ∈ Bα(X, [0, 1]) such that Ai = h −1 i (0) for i = 1, 2. We set h(x) = h1(x) h1(x) + h2(x) for all x ∈ X. It is easy to verify that h ∈ Bα(X, [0, 1]) and Ai = h −1(i − 1), i = 1, 2. Consider a continuous mapping λ : Y ×Y ×[0, 1] → Y such that λ(y, z, 0) = y and λ(y, z, 1) = z for all y, z ∈ Y . Let f(x) = λ(f1(x), f2(x), h(x)) for every x ∈ X. Clearly, f ∈ Bα(X, Y ). If x ∈ A1, then f(x) = λ(f1(x), f2(x), 0) = f1(x). If x ∈ A2, then f(x) = λ(f1(x), f2(x), 1) = f2(x). Assume that the lemma is true for all 2 ≤ k < n and let k = n. According to our assumption, there exists a mapping g ∈ Bα(X, Y ) such that g|Ai = fi for all 1 ≤ i < n. Since A = n−1 ⋃ i=1 Ai and An are disjoint sets which belong to the α’th functionally multiplicative class in X, by the assumption, there is a mapping f ∈ Bα(X, Y ) with f|A = g and f|Fn = fn. Then f|Fi = fi for every 1 ≤ i ≤ n. � Let 0 ≤ α < ω1. We say that a mapping f : X → Y is of the (functional) α-th Lebesgue class, f ∈ Hα(X, Y ) (f ∈ H ∗ α(X, Y )), if the preimage f −1(V ) belongs to the α’th (functionally) additive class in X for any open set V ⊆ Y . Clearly, Hα(X, Y ) = H ∗ α(X, Y ) for any perfectly normal space X. The following statement is well-known, but we present a proof here for con- venience of the reader. Lemma 2.3. Let X and Y be topological spaces, (fk) ∞ k=1 a sequence of map- pings fk : X → Y which is pointwise convergent to a mapping f : X → Y , F ⊆ Y be a closed set such that F = ∞ ⋂ n=1 V n, where (Vn) ∞ n=1 is a sequence of open sets in Y such that Vn+1 ⊆ Vn for all n ∈ N. Then (2.1) f−1(F) = ∞ ⋂ n=1 ∞ ⋃ k=n f −1 k (Vn). Classification of separately continuous mappings 171 Proof. Let x ∈ f−1(F) and n ∈ N. Taking into account that Vn is an open neighborhood of f(x) and lim k→∞ fk(x) = f(x), we obtain that there is k ≥ n such that fk(x) ∈ Vn. Now let x belong to the right-hand side of (2.1), i.e. for every n ∈ N there exists a number k ≥ n such that fk(x) ∈ Vn. Suppose f(x) 6∈ F . Then there exists n ∈ N such that f(x) 6∈ Vn. Since U = X \ Vn is a neighborhood of f(x), there exists k0 such that fk(x) ∈ U for all k ≥ k0. In particular, fk(x) ∈ U for k = max{k0, n}. But then fk(x) 6∈ Vn, a contradiction. Hence, x ∈ f −1(F). � Lemma 2.4. Let X be a topological space, Y a perfectly normal space and 0 ≤ α < ω1. Then Bα(X, Y ) ⊆ H ∗ α(X, Y ) if α is finite, and Bα(X, Y ) ⊆ H∗α+1(X, Y ) if α is infinite. Proof. Let f ∈ Bα(X, Y ). Fix an arbitrary closed set F ⊆ Y . Since Y is perfectly normal, there exists a sequence of open sets Vn ⊆ Y such that Vn+1 ⊆ Vn and F = ∞ ⋂ n=1 V n. Moreover, there exists a sequence of mappings fk : X → Y of Baire classes < α which is pointwise convergent to f on X. By Lemma 2.3, equality (2.1) holds. Now put An = ∞ ⋃ k=n f −1 k (Vn). If α = 0, then f is continuous and f−1(F) is a zero set in X, since F is a zero set in Y . Suppose the assertion of the lemma is true for all finite ordinals 1 ≤ ξ < α. We show that it is true for α. Remark that fk ∈ Bα−1(X, Y ) for every k ≥ 1. By assumption, fk ∈ H ∗ α−1(X, Y ) for every k ∈ N. Then An is of the functionally additive class α − 1. Therefore, f−1(F) belongs to the α’th functionally multiplicative class. Assume the assertion of the lemma is true for all ordinals ω0 ≤ ξ < α. For all k ∈ N we choose αk < α such that fk ∈ Bαk (X, Y ) for every k ≥ 1. The preimage f−1 k (Vn), being of the (αk + 1)’th functionally additive class, belongs to the α’th functionally additive class for all k, n ∈ N, provided αk + 1 ≤ α. Then An is of the α’th functionally additive class, hence, f −1(F) belongs to the (α + 1)’th functionally multiplicative class. � Recall that a family A = (Ai : i ∈ I) of sets Ai refines a family B = (Bj : j ∈ J) of sets Bj if for every i ∈ I there exists j ∈ J such that Ai ⊆ Bj. We write in this case A � B. 3. PP-spaces and their properties Definition 3.1. A topological space X is said to be a (strong) PP-space if (for every dense set D in X) there exist a sequence ((ϕi,n : i ∈ In)) ∞ n=1 of locally finite partitions of unity on X and a sequence ((xi,n : i ∈ In)) ∞ n=1 of families of points of X (of D) such that (3.1) (∀x ∈ X)((∀n ∈ N x ∈ suppϕin,n) =⇒ (xin,n → x)) Remark that Definition 3.1 is equivalent to the following one. 172 O. Karlova Definition 3.2. A topological space X is a (strong) PP-space if (for every dense set D in X) there exist a sequence ((Ui,n : i ∈ In)) ∞ n=1 of locally finite covers of X by co-zero sets Ui,n and a sequence ((xi,n : i ∈ In)) ∞ n=1 of families of points of X (of D) such that (3.2) (∀x ∈ X)((∀n ∈ N x ∈ Uin,n) =⇒ (xin,n → x)) Clearly, every strong PP-space is a PP-space. Proposition 3.3. Every metrizable space is a strong PP-space. Proof. Let X be a metrizable space and d a metric on X which generates its topology. Fix an arbitrary dense set D in X. For every n ∈ N let Bn be a cover of X by open balls of diameter 1 n . Since X is paracompact, for every n there exists a locally finite cover Un = (Ui,n : i ∈ In) of X by open sets Ui,n such that Un � Bn. Notice that each Ui,n is a co-zero set. Choose a point xi,n ∈ D ∩Ui,n for all n ∈ N and i ∈ In. Let x ∈ X and let U be an arbitrary neighborhood of x. Then there is n0 ∈ N such that B(x, 1 n ) ⊆ U for all n ≥ n0. Fix n ≥ n0 and take i ∈ In such that x ∈ Ui,n. Since diam Ui,n ≤ 1 n , d(x, xi,n) ≤ 1 n , consequently, xi,n ∈ U. � Example 3.4. The Sorgenfrey line L is a strong PP-space which is not metri- zable. Proof. Recall that the Sorgenfrey line is the real line R endowed with the topology generated by the base consisting of all semi-intervals [a, b), where a < b (see [3, Example 1.2.2]). Let D ⊆ L be a dense set. For any n ∈ N and i ∈ Z by ϕi,n we denote the characteristic function of [i−1 n , i n ) and choose a point xi,n ∈ [ i n , i+1 n )∩D. Then the sequences ( (ϕi,n : i ∈ In) )∞ n=1 and ( (xi,n : i ∈ In) )∞ n=1 satisfy (3.1). � Proposition 3.5. Every σ-metrizable paracompact space is a PP-space. Proof. Let X = ∞ ⋃ n=1 Xn, where (Xn) ∞ n=1 is an increasing sequence of closed metrizable subspaces, and let d1 be a metric on X1 which generates its topology. According to Hausdorff’s theorem [3, p. 297] we can extend the metric d1 to a metric d2 on X2. Further, we extend the metric d2 to a metric d3 on X3. Repeating this process, we obtain a sequence (dn) ∞ n=1 of metrics dn on Xn such that dn+1|Xn = dn for every n ∈ N. We define a function d : X 2 → R by setting d(x, y) = dn(x, y) for (x, y) ∈ X 2 n. Fix n ∈ N and m ≥ n. Let Bn,m be a cover of Xm by d-open balls of diameter 1 n . For every B ∈ Bn,m there exists an open set VB in X such that VB ∩ Xm = B. Let Vn,m = {VB : B ∈ Bn,m} and Un = ∞ ⋃ m=1 Vn,m. Then Un is an open cover of X for every n ∈ N. Since X is paracompact, for every n ∈ N there exists a locally finite partition of unity (hi,n : i ∈ In) on X subordinated to Un. For every n ∈ N and i ∈ In we choose xi,n ∈ Xk(i,n) ∩ supp hi,n, where k(i, n) = min{m ∈ N : Xm ∩ supp hi,n 6= Ø}. Classification of separately continuous mappings 173 Now fix x ∈ X. Let (in)n=1 be a sequence of indexes in ∈ In such that x ∈ supp hin,n. We choose m ∈ N such that x ∈ Xm. It is easy to see that k(in, n) ≤ m for every n ∈ N. Then xin,n ∈ Xm. Since dm(xin,n, x) ≤ diam supp hin,n ≤ 1 n , xin,n → x in Xm. Therefore, xin,n → x in X. � Denote by R∞ the collection of all sequences with a finite support, i.e. se- quences of the form (ξ1, ξ2, . . . , ξn, 0, 0, . . . ), where ξ1, ξ2, . . . , ξn ∈ R. Clearly, R ∞ is a linear subspace of the space RN of all sequences. Denote by E the set of all sequences e = (εn) ∞ n=1 of positive reals εn and let Ue = {x = (ξn) ∞ n=1 ∈ R ∞ : (∀n ∈ N)(|ξn| ≤ εn)}. We consider on R∞ the topology in which the system U0 = {Ue : e ∈ E} forms the base of neighborhoods of zero. Then R∞ equipped with this topology is a locally convex σ-metrizable paracompact space which is not a first countable space, consequently, non-metrizable. Example 3.6. The space R∞ is a PP-space which is not a strong PP-space. Proof. Remark that R∞ is a PP-space by Proposition 3.5. We show that R∞ is not a strong PP-space. Indeed, let An = {(ξ1, ξ2, . . . , ξn, 0, 0, . . . ) : |ξk| ≤ 1 n (∀1 ≤ k ≤ n)}, D = ∞ ⋃ m=1 m ⋂ n=1 (R∞ \ An). Then D is dense in R∞, but there is no sequence in D which converges to x = (0, 0, 0, . . .) ∈ R∞. Hence, R∞ is not a strong PP-space. � 4. The Lebesgue classification The following result is an analog of theorems of K. Kuratowski [11] and D. Montgomery [14] who proved that every separately continuous function, defined on a product of a metrizable space and a topological space and with values in a metrizable space, belongs to the first Baire class. Theorem 4.1. Let X be a strong PP-space, Y a topological space, Z a perfectly normal space and 0 ≤ α < ω1. Then CH∗α(X × Y, Z) ⊆ H ∗ α+1(X × Y, Z). Proof. Let f ∈ CH∗α(X ×Y, Z). Then for the set XH∗α(f) there exist a sequence (Un) ∞ n=1 of locally finite covers Un = (Ui,n : i ∈ In) of X by co-zero sets Ui,n and a sequence ((xi,n : i ∈ In)) ∞ n=1 of families of points of the set XH∗α(f) satisfying condition (3.2). We choose an arbitrary closed set F ⊆ Z. Since Z is perfectly normal, F = ∞ ⋂ m=1 Gm, where Gm are open sets in Z such that Gm+1 ⊆ Gm for every 174 O. Karlova m ∈ N. Let us verify that the equality (4.1) f−1(F) = ∞ ⋂ m=1 ∞ ⋃ n≥m ⋃ i∈In Ui,n × (f xi,n)−1(Gm). holds. Indeed, let (x0, y0) ∈ f −1(F). Then f(x0, y0) ∈ Gm for every m ∈ N. Fix any m ∈ N. Since Vm = f −1 y0 (Gm) is an open neighborhood of x0, there exists a number n0 ≥ m such that for all n ≥ n0 and i ∈ In the inclusion xi,n ∈ Vm holds whenever x0 ∈ Ui,n. We choose i0 ∈ In0 such that x0 ∈ Ui0,n0. Then f(xi0,n0, y0) ∈ Gm. Hence, (x0, y0) belongs to the right-hand side of (4.1). Conversely, let (x0, y0) belong to the right-hand side of (4.1). Fix m ∈ N. We choose sequences (nk) ∞ k=1, (mk) ∞ k=1 of numbers nk, mk ∈ N and a sequence (ik) ∞ k=1 of indexes ik ∈ Ink such that m = m1 ≤ n1 < m2 ≤ n2 < · · · < mk ≤ nk < . . . , x0 ∈ Uik,nk and f(xik,nk, y0) ∈ Gmk ⊆ Gm for every k ∈ N. Since lim k→∞ xik,nk = x0 and the mapping f is continuous with respect to the first variable, lim k→∞ f(xik,nk, y0) = f(x0, y0). Therefore, f(x0, y0) ∈ Gm for every m ∈ N. Hence, (x0, y0) belongs to the left-hand side of (4.1). Since fxi,n ∈ H∗α(Y, Z), the sets (f xi,n)−1(Gm) are of the functionally addi- tive class α in Y . Moreover, all Ui,n are co-zero sets in X, consequently, by [6, Theorem 1.5] the set En = ⋃ i∈In Ui,n × (f xi,n)−1(Gm) belongs to the α’th func- tionally additive class for every n. Therefore, ⋃ n≥m En is of the α’th functionally additive class. Hence, f−1(F) is of the (α + 1)’th functionally multiplicative class in X × Y . � Definition 4.2. We say that a topological space X has the (strong) L-property or is a (strong) L-space, if for every topological space Y every (nearly vertically) separately continuous function f : X × Y → R is of the first Lebesgue class. According to Theorem 4.1 every strong PP-space has the strong L-property. Proposition 4.3. Let X be a completely regular strong L-space. Then for any dense set A ⊆ X and a point x0 ∈ X there exists a countable dense set A0 ⊆ A such that x0 ∈ A0. Proof. Fix an arbitrary everywhere dense set A ⊆ X and a point x0 ∈ A. Let Y be the space of all real-valued continuous functions on X, endowed with the topology of pointwise convergence on A. Since the evaluation function e : X × Y → R, e(x, y) = y(x), is nearly vertically separately continuous, e ∈ H1(X × Y, R). Then B = e −1(0) is Gδ-set in X × Y . Hence, B0 = {y ∈ Y : y(x0) = 0} is a Gδ-set in Y . We set y0 ≡ 0 and choose a sequence (Vn) ∞ n=1 of basic neighborhoods of y0 in Y such that ∞ ⋂ n=1 Vn ⊆ B0. For every n there Classification of separately continuous mappings 175 exist a finite set {xi,n : i ∈ In} of X and εn > 0 such that Vn = {y ∈ Y : max i∈In |y(xi,n)| < εn}. Let A0 = ⋃ n∈N ⋃ i∈In {xi,n}. Take an open neighborhood U of x0 in X and suppose that U ∩ A0 = Ø. Since X is completely regular and x0 6∈ X \ U, there exists a continuous function y : X → R such that y(x0) = 1 and y(X \ U) ⊆ {0}. Then y ∈ ∞ ⋂ n=1 Vn, but y 6∈ B0, a contradiction. Therefore, U ∩ A0 6= Ø, and x0 ∈ A0. � 5. Baire classification and σ-metrizable spaces We recall that a topological space Y is B-favorable for a space X, if H1(X, Y ) ⊆ B1(X, Y ) (see [10]). Definition 5.1. Let 0 ≤ α < ω1. A topological space Y is called weakly Bα-favorable for a space X, if H ∗ α(X, Y ) ⊆ Bα(X, Y ). Clearly, every B-favorable space is weakly B1-favorable. Proposition 5.2. Let 0 ≤ α < ω1, X a topological space, Y = ∞ ⋃ n=1 Yn a contractible space, f : X → Y a mapping, (Xn) ∞ n=1 a sequence of sets of the α’th functionally additive class such that X = ∞ ⋃ n=1 Xn and f(Xn) ⊆ Yn for every n ∈ N. If one of the following conditions holds (i) Yn is a nonempty weakly Bα-favorable space for X for all n and f ∈ H∗α(X, Y ), or (ii) α > 0 and for every n there exists a mapping fn ∈ Bα(X, Yn) such that fn|Xn = f|Xn, then f ∈ Bα(X, Y ). Proof. If α = 0 then the statement is obvious in case (i). Let α > 0. By [6, Lemma 2.1] there exists a sequence (En) ∞ n=1 of disjoint functionally ambiguous sets of the α’th class such that En ⊆ Xn and X = ∞ ⋃ n=1 En. In case (i) for every n we choose a point yn ∈ Yn and let fn(x) = { f(x), if x ∈ En, yn, if x ∈ X \ En. Since f ∈ H∗α(X, Y ) and En is functionally ambiguous set of the α’th class, fn ∈ H ∗ α(X, Yn). Then fn ∈ Bα(X, Yn) provided Yn is weakly Bα-favorable for X. For every n there exists a sequence of mappings gn,m : X → Yn of classes < α such that gn,m(x) → m→∞ fn(x) for every x ∈ X. In particular, lim m→∞ gn,m(x) = 176 O. Karlova f(x) on En. Since En is of the α-th functionally additive class, En = ∞ ⋃ m=1 Bn,m, where (Bn,m) ∞ m=1 is an increasing sequence of sets of functionally additive classes < α. Let Fn,m = Ø if n > m, and let Fn,m = Bn,m if n ≤ m. According to Lemma 2.2, for every m ∈ N there exists a mapping gm : X → Y of a class < α such that gm|Fn,m = gn,m, since the system {Fn,m : n ∈ N} is finite for every m ∈ N. It remains to prove that gm(x) → f(x) on X. Let x ∈ X. We choose a number n ∈ N such that x ∈ En. Since the sequence (Fn,m) ∞ m=1 is increas- ing, there exists a number m0 such that x ∈ Fn,m for all m ≥ m0. Then gm(x) = gn,m(x) for all m ≥ m0. Hence, lim m→∞ gm(x) = lim m→∞ gn,m(x) = f(x). Therefore, f ∈ Bα(X, Y ). � Definition 5.3. Let {Xn : n ∈ N} be a cover of a topological space X. We say that (X, (Xn) ∞ n=1) has the property (∗) if for every convergent sequence (xk) ∞ k=1 in X there exists a number n such that {xk : k ∈ N} ⊆ Xn. Proposition 5.4. Let 0 ≤ α < ω1, X a strong PP-space, Y a topological space, (Z, (Zn) ∞ n=1) have the property (∗), let Zn be closed in Z (and let Zn be a zero-set in Z if α = 0) for every n ∈ N, and f ∈ CBα(X × Y, Z). Then there exists a sequence (Bn) ∞ n=1 of sets of the α’th /(α + 1)’th/ functionally multiplicative class in X × Y , if α is finite /infinite/, such that ∞ ⋃ n=1 Bn = X × Y and f(Bn) ⊆ Zn for every n ∈ N. Proof. Since XBα(f) is dense in X, there exists a sequence (Um = (Ui,m : i ∈ Im)) ∞ m=1 of locally finite co-zero covers of X and a sequence ((xi,m : i ∈ Im)) ∞ m=1 of families of points of XBα(f) such that condition (3.2) holds. In accordance with [16, Proposition 3.2] there exists a pseudo-metric on X such that all the set Ui,m are co-zero with respect to this pseudo-metric. Denote by T the topology on X generated by the pseudo-metric. Obviously, the topology T is weaker than the initial one. Using the paracompactness of (X, T ), for every m we choose a locally finite open cover Vm = (Vs,m : s ∈ Sm) which refines Um. By [3, Lemma 1.5.6], for every m there exists a locally finite closed cover (Fs,m : s ∈ Sm) of (X, T ) such that Fs,m ⊆ Vs,m for every s ∈ Sm. Now for every s ∈ Sm we choose i(s) ∈ Im such that Fs,m ⊆ Ui(s),m. For all m, n ∈ N and s ∈ Sm let As,m,n = (f xi(s),m)−1(Zn), Bm,n = ⋃ s∈Sm (Fs,m × As,m,n), Bn = ∞ ⋂ m=1 Bm,n. Since f is of the α’th Baire class with respect to the second variable, for every n the set As,m,n belongs to the α’th functionally multiplicative class /α+1/ in Y for all m ∈ N and s ∈ Sm, if α is finite /infinite/ by Lemma 2.4. According to [6, Proposition 1.4] the set Bm,n is of the α’th /(α + 1)’th/ functionally Classification of separately continuous mappings 177 multiplicative class in (X, T ) × Y . Then the set Bn is of the α’th /(α + 1)’th/ functionally multiplicative class in (X, T )× Y , and, consequently, in X × Y for every n. We prove that f(Bn) ⊆ Zn for every n. To do this, fix n ∈ N and (x, y) ∈ Bn. We choose a sequence (sm) ∞ m=1 such that x ∈ Fm,sm ⊆ Um,i(sm) and f(xm,i(sm), y) ∈ Zn. Then xm,i(sm) → m→∞ x. Since f is continuous with respect to the first variable, f(xm,i(sm), y) → m→∞ f(x, y). The set Zn is closed, then f(x, y) ∈ Zn. Now we show that ∞ ⋃ n=1 Bn = X ×Y . Let (x, y) ∈ X ×Y . Then there exists a sequence (sm) ∞ m=1 such that x ∈ Fm,sm ⊆ Um,i(sm) and f(xm,i(sm), y) → m→∞ f(x, y). Since (Z, (Zn) ∞ n=1) satisfies (∗), there is a number n such that {f(xm,im, y) : m ∈ N} is contained in Zn, i.e. y ∈ Am,n,i for every m ∈ N. Hence, (x, y) ∈ Bn. � Theorem 5.5. Let X be a strong PP-space, Y a topological space, {Zn : n ∈ N} a closed cover of a contractible perfectly normal space Z, let (Z, (Zn) ∞ n=1) satisfy (∗) and Zn be weakly B1-favorable for X × Y for every n ∈ N. Then CC(X × Y, Z) ⊆ B1(X × Y, Z). Proof. Let f ∈ CC(X × Y, Z). In accordance with Theorem 4.1, f ∈ H∗1 (X × Y, Z). Moreover, Proposition 5.4 implies that there exists a sequence of zero- sets Bn ⊆ X × Y such that ∞ ⋃ n=1 Bn = X × Y and f(Bn) ⊆ Zn for every n ∈ N. Since for every n the set Bn is an F ∗ σ -set and H ∗ 1 (X × Y, Zn) ⊆ B1(X × Y, Zn), f ∈ B1(X × Y, Z) by Proposition 5.2. � Definition 5.6. A topological space X is called strongly σ-metrizable, if it is σ- metrizable with a stratification (Xn) ∞ n=1 and (X, (Xn) ∞ n=1) has the property (∗). Taking into account that every regular strongly σ-metrizable space with metrizable separable stratification is perfectly normal (see [13, Corollary 4.1.6]) and every metrizable separable arcwise connected and locally arcwise connected space is weakly Bα-favorable for any topological space X for all 0 ≤ α < ω1 [7, Theorem 3.3.5], we immediately obtain the following corollary of Theorem 5.5. Corollary 5.7. Let X be a strong PP-space, Y a topological space and Z a contractible regular strongly σ-metrizable space with a stratification (Zn) ∞ n=1, where Zn is a metrizable separable arcwise connected and locally arcwise con- nected space for every n ∈ N. Then CC(X × Y, Z) ⊆ B1(X × Y, Z). 178 O. Karlova References [1] T. Banakh, (Metrically) quarter-stratifiable spaces and their applications, Math. Stud. 18, no. 1 (2002), 10–28. [2] M. Burke, Borel measurability of separately continuous functions, Topology Appl. 129, no. 1 (2003), 29–65. [3] R. Engelking, General Topology. 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