Borel Sets and Functions in Topological Spaces

Home , Baire function, Borel hierarchy, Borel set

BOREL SETS AND FUNCTIONS IN TOPOLOGICAL SPACES

JIRˇ´I SPURNY´

Abstract. We present a construction of the Borel hierarchy in general topological spaces and its relation to Baire hierarchy. We deﬁne mappings of Borel class α, prove the validity of the Lebesgue–Hausdorﬀ–Banach characterization for them and show their relation to Baire classes of mappings on compact spaces. Obtained result are used for studying Baire and Borel order of compact spaces, answering thus one part of the question asked by R.D. Mauldin. We present several examples showing some natural limits of our results in non-compact spaces.

1. Introduction The Borel hierarchy is a classical topic deeply studied within the framework of metrizable spaces, we refer the reader to [1], [31], [18], [6], [46] and [28] for detailed surveys of results in separable metrizable spaces and [48], [13], [14], [22] and [21] for results in non-separable metrizable spaces. On the other hand, the question how to develop a satisfactory theory of Borel classes in general topological spaces have been studied much less extensively (see e.g. [8], [19], [20], [16], [17], [40], [41] and [23]). The aim of our paper is to present an attempt in this direction. As one expects, we often use well known methods. We have tried to quote all sources known to us that are needed and/or closely related to our work. However, because of the vast amount of work in this area we have not attempted to collect all the references connected with this topic. Since it requires some effort to introduce all notions, we postpone the definitions and results to the relevant sections and just briefly describe the content of the paper. We start by the introduction of Borel classes defined in an abstract way from a general family of sets (see Definition 2.1). Then we introduce abstract classes of mappings defined as pointwise limits from a given family of mappings and prove the Lebesgue–Hausdorff–Banach characterization of these mappings via the preimages of open sets (see Theorem 2.7). The main Section 3 introduces classes of Baire, Borel and resolvable sets in topological spaces. We show their relation to the classical Borel hierarchy in metrizable spaces (Proposition 3.4). The main result formulated in Theorem 3.6 shows coin- cidence of all defined classes for Baire sets in compact spaces. A key method used in the proof of Theorem 3.6 is the selection lemma [23, Lemma 1]. We finish the

2000 Mathematics Subject Classification. 54H05, 28A05. Key words and phrases. Baire sets, Borel sets, H–sets, Baire functions, Borel functions, Baire and Borel order, scattered–K–analytic, Cechˇ analytic and K–analytic spaces, functions of the first Baire and Borel class. The work is a part of the research project MSM 0021620839 financed by MSMT and partly supported by the grants GA CRˇ 201/06/0018, and GA CRˇ 201/07/0388. 1 2 JIRˇÍ SPURNY´ section by presenting a result on “absoluteness” of Baire classes in Theorem 3.7 (cf. [23, Theorem 5]). Results of Section 3 are used in Section 4 for studying order of Baire, Borel and resolvable sets in topological spaces. Theorem 4.2 answers one part of the question asked by R.D. Mauldin in [34, Question, p. 440] and [35, Problem, p. 395]. The next section starts by proving the Lebesgue–Hausdorff–Banach characterization of mappings on compact spaces (Theorem 5.2). Then we prove in Corollary 5.6 the equality between Baire measurable mappings and mappings of some Baire class for mappings with values in arcwise connected locally arcwise connected metric spaces and provide several characterizations of mappings of the first Baire class on compact spaces (Theorem 5.8). Several examples and questions finish the paper. Let us recall several definitions. If X is a set and F is a family of its subsets, then F is a sublattice, if ∅, X ∈ F and F is closed with respect to finite unions and intersections. The family F is an algebra if F is a sublattice that is closed with respect to complements. If F is a family of sets in a set X, we write Fσ (respectively Fδ) for all countable unions (respectively intersections) of sets from F. We write χA for the characteristic function of a set A and f|A for the restriction of a function f on A. If f : X → Y is a mapping from X to a topological space Y , we say that f is F–measurable, if f −1(U) ∈F for each open U ⊂ Y . All topological spaces are considered to be Tychonoff (see [4, p. 39].

2. Generation of sets and functions Deﬁnition 2.1. If F is a family of sets in a set X, we deﬁne Borel classes generated by F as follows: Let Σ1(F) = F, Π1(F) = {X \ F : F ∈ F}, and for α ∈ (1, ω1), let

Σα(F) = Πβ(F) σ β<α [ and

Πα(F) = Σβ(F) . δ β<α [ The family Σα(F) is termed the sets of additive class α, the family Πα(F) is called the sets of multiplicative class α. The sets in ∆α(F) = Σα(F) ∩ Πα(F) are the sets of ambiguous class α. Remark 2.2. The deﬁnition and notation of abstract Borel classes is chosen in such a way that it coincides with the deﬁnition of Borel sets in metrizable spaces, see [28, Section 11.B] and Proposition 3.4. The following proposition collects facts on abstract Borel classes that one expects to hold (see [28, Section 11.B, Exercise 22.17 and Proposition 22.15] or [31, §26]). Proposition 2.3. Let F be an algebra of sets in a set X. Then the following assertions hold:

(a) Σα(F) ∪ Πα(F) ⊂ ∆β(F), 0 < α < β < ω1, (b) a set A ⊂ X is in Σα(F) if and only if X \ A ∈ Πα(F), α ∈ (0, ω1), (c) the family Σα(F) is stable with respect to countable unions and ﬁnite intersections, Πα(F) is stable with respect to ﬁnite unions and countable intersections and ∆α(F) is an algebra, α ∈ (1, ω1), BOREL SETS AND FUNCTIONS IN TOPOLOGICAL SPACES 3

(d) α<ω1 Σα(F) = α<ω1 Πα(F) is the σ–algebra generated by F, (e) Σα(F) = (∆α(F))σ, α ∈ (1, ω ), S S 1 (f) for each α ∈ (1, ω1), Σα(F) has the generalized reduction property, that is, ′ if An ∈ Σα(F), n ∈ N, then there exist pairwise disjoint sets An ∈ Σα(F), ′ ∞ ′ ∞ n ∈ N, such that An ⊂ An and n=1 An = n=1 An, (g) if α ∈ (1, ω1) and A ∈ ∆α+1(F), then there exists a sequence {An} of sets S S such that An ∈ ∆α(F), n ∈ N, and χA = lim χAn , (h) if α ∈ (1, ω1) is limit and A ∈ ∆α+1(F), then there exists a sequence {An} of sets such that An ∈ β<α ∆β(F), n ∈ N, and χA = lim χAn . Proof. Assertions (a), (b) and (c)S follow by a straightforward transﬁnite induction, (d) follows from (a) and (c), and (e) follows from (a). For the proof of (f), let An ∈ ∞ Σα(F), n ∈ N, be given. Using (e) we write An = k=1 Ank, where Ank ∈ ∆α(F). ′ We enumerate the family {Ank : n, k ∈ N} as {B } and deﬁne j S ′ ′ ′ ′ B1 = B1, Bj = Bj \ (B1 ∪ · · · ∪ Bj−1), j ≥ 2.

Then the sets Bj are contained in ∆α(F). We set

′ A1 = {Bj : Bj ⊂ A1} and ′ [ ′ ′ An = {Bj : Bj ⊂ An,Bj * A1 ∪ · · · ∪ An−1}, n ≥ 2. ′ [ ′ ′ Then An ⊂ An and An ∈ Σα(F), n ∈ N. Moreover, {An : n ∈ N} is a disjoint ∞ family whose union equals to n=1 An. To verify (g), let A ∈ ∆α+1(F) for some α ∈ (1, ω1) be given. By the definition, ∞ ∞ S A = n=1 Bn = n=1 Cn, where Bn ∈ Πα(F), Cn ∈ Σα(F), n ∈ N. We fix n ∈ N and use (f) for the couple {X \ Bn, Cn} to find a set An ∈ ∆α(F) such that S T Bn ⊂ An ⊂ Cn.

Then lim χAn = χA, as required. For the proof of (h), let α be a countable limit ordinal and A ∈ ∆α+1(F). We ∞ ∞ write A = n=1 Bn = n=1 Cn, where Bn ∈ Πα(F), Cn ∈ Σα(F), n ∈ N. By (a), there exists ordinals αnk < α and sets Bnk, Cnk ∈ ∆αnk (F) so that S T ∞ ∞ Bn = Bnk and Cn = Cnk, n, k ∈ N. k k \=1 [=1 Without loss of generality we may assume that the sequences {Bn}, {Cnk} are increasing and the sequences {Cn}, {Bnk} are decreasing. Then the sets n p An = (Bpn ∩ Cjn), n ∈ N, p=1 j=1 [ \ are contained in ∆αn (F) for some αn < α and lim χAn = χA (see [34, proof of Theorem 2.1]). This concludes the proof.

Definition 2.4. If Φ is a family of mappings from a set X to a topological space Y , inductively we define Baire classes generated by Φ as follows: Let Φ0 = Φ and for each countable ordinal α ∈ (0, ω1), let Φα be the family of all pointwise limits of sequences from β<α Φβ. S 4 JIRˇÍ SPURNY´

Remark 2.5. Later on (see Theorem 2.7 and Deﬁnition 5.1), it will be sometimes convenient to denote the starting family of the inductive deﬁnition as Φ1. More precisely, we start from a family denoted as Φ1 and then Φα consists of all pointwise limits of sequences from 1≤β<α Φβ, α ∈ (1, ω1). The purpose of this convention is that we want to start the generation of mappings between topological spaces from “Baire–one” mappings. S The proof of Theorem 2.7 below closely follows the proof of [28, Theorem 24.3], however we include the following lemma. Lemma 2.6. Let F be an algebra of sets in a set X and let 0 <ε<ε′. Let f, g : X → Y be mappings from a set X to a separable metric space (Y, ρ) such that ρ(f(x), g(x)) < ε, x ∈ X. Let fn, gn : X → Y , n ∈ N, be Σγn (F)–measurable mappings such that fn → f, gn → g and γn ∈ (1, ω1).

Then there exist Σγn (F)–measurable mappings hn : X → Y , n ∈ N, so that ′ hn → g and ρ(fn(x),hn(x)) < ε for each x ∈ X and n ∈ N. Proof. Given the objects as in the premise, for n ∈ N we set 1 ′ 2 An = {x ∈ X : ρ(fn(x), gn(x)) < ε }, An = {x ∈ X : ρ(fn(x), gn(x)) > ε}. 1 2 By separability of the space Y we get that the sets An, An are in Σγn (F). Indeed, we select countably many open sets Uk, Vk, k ∈ N, in Y such that ∞ ′ (Uk × Vk) = {(y1,y2) ∈ Y × Y : ρ(y1,y2) < ε }. k [=1 Hence ∞ 1 −1 −1 An = (fn (Uk) ∩ gn (Vk)) k [=1 2 is contained in Σγn (F). Similarly we argue for the set An. 1 2 Thus we may use Proposition 2.3(f) to ﬁnd sets Bn,Bn ∈ Σγn (F) such that 1 1 2 2 1 2 Bn ⊂ An, Bn ⊂ An, and Bn ∪ Bn = X, n ∈ N. By setting 1 gn, on Bn, hn = 2 n ∈ N, (fn, on Bn, we get Σγn (F)–measurable mappings satisfying our requirements.

Theorem 2.7. Let F be an algebra of sets in a set X and let Y be a separable metrizable space. Let Φ1 stand for the family of all Σ2(F)–measurable mappings from X to Y and, for α ∈ (1, ω1), let Φα be deﬁned from Φ1 as in Deﬁnition 2.4. Then, for each α ∈ (0, ω1) and f : X → Y , the following assertions are equivalent:

(i) f ∈ Φα, (ii) f is Σα+1(F)–measurable. Proof. We prove (i) =⇒ (ii) by transﬁnite induction. If α = 1, the assertion holds by the very deﬁnition. We assume its validity for all β < α, where α ∈ (1, ω1). If BOREL SETS AND FUNCTIONS IN TOPOLOGICAL SPACES 5

f ∈ Φα, then f = lim fn, where fn ∈ Φαn for some αn < α, n ∈ N. Given an open ∞ set U ⊂ Y , we write U = m=1 Um, where the sets Um are open. Then ∞ ∞ ∞ S−1 −1 f (U) = fk (Um), m=1 n=1 k n [ [ \= −1 and the sets fk (Um) ∈ Παk+1(F) according to the inductive assumption. Thus −1 f (U) ∈ Σα+1(F). For the proof of (ii) =⇒ (i) we use transfinite induction again. The case α = 1 follows from the definition. Assume that α ∈ (1, ω1) and the assertion hold for all β < α. Let f : X → Y be a Σα+1(F)–measurable function. We assume first that f : X → Y has only finitely many values, that is, there exists pairwise disjoint sets Ai ⊂ X and points yi ∈ Y , i = 1,...,n, such that Ai ∈ ∆α+1(F) and f = yi on Ai, i = 1,...,n. ′ ′ Assume first that α is isolated, that is, α = α + 1 for some α ∈ [1, ω1). Ac- k ∞ cording to Proposition 2.3(g), there exist sequences {Ai }k=1 of sets in ∆α(F), k k k i = 1,...,n, such that χAi → χAi . We may assume that {A1 ,...,An} is a dis- k k k joint family for each k ∈ N. (Otherwise we would take B1 = A1 and Bj = k k k Aj \ (A1 ∪ · · · ∪ Aj−1), j = 2,...,n.) Then k fk(x) = yi, x ∈ Ai , i = 1,...,n, k ∈ N, are Σα(F)–measurable mappings that converge pointwise to f. According to the inductive assumption, the mappings fk are contained in Φα′ and so f ∈ Φα. Assume that α is limit. In this case we use Proposition 2.3(h) and find sequences k ∞ {Ai }k=1 of sets in ∆αk (F), i = 1,...,n, where αk < α for every k ∈ N, such that k χAi → χAi , i = 1,...,n. The rest of the argument is analogous as above. Assume now that f : X → Y is an arbitrary Σα+1(F)–measurable mapping. Since Y is separable metrizable, we can fix on Y a compatible metric ρ such that k k k (Y, ρ) is precompact. Hence we may find finite sets Y = {y1 ,...,ynk } ⊂ Y , k ∈ N, k k+1 nk k −k so that Y ⊂ Y and Y ⊂ i=1 U(yi , 2 ). (Here U(y, r) stand for the open ball centered at y with radius r.) −1 Sk −k We fix k ∈ N. Then {f (U(yi , 2 )) : i = 1,...,nk} is a cover of X consisting k of sets in Σα+1(F). Using Proposition 2.3(f) we find a partition {Ai : i = 1,...,nk} k −1 k −k of X consisting of sets in ∆α+1(F) so that Ai ⊂ f (U(yi , 2 )), i = 1,...,nk. Then the function k k f (x) = yi, x ∈ Ai , i = 1,...,nk, is Σα+1(F)–measurable. According to the reasoning above, we may find mappings k k k fn ∈ Φαnk so that fn → f and αnk < α. By implication (i) =⇒ (ii), k (1) fn ∈ Σαnk+1(F), n, k ∈ N. If x ∈ X, from ρ(f k(x), f(x)) < 2−k it follows that ρ(f k(x), f k+1(x)) < 2−k+1. k k Now we employ Lemma 2.6 to replace {fn } by {hn} in such a way that 1 1 • hn = fn, n ∈ N, k k+1 −k+2 • ρ(hn(x),hn (x)) < 2 , x ∈ X, n, k ∈ N, k • hn is Σγnk+1(F)–measurable for some γnk < α, n, k ∈ N. To do this, we proceed inductively. First we set 1 1 hn = fn, n ∈ N, 6 JIRˇÍ SPURNY´

j and having hn : X → Y deﬁned for all n ∈ N and j = 1,...,k−1, we use Lemma 2.6 k to get mappings hn : X → Y , n ∈ N, so that k−1 k −(k−1)+2 ρ(hn (x),hn(x)) < 2 , x ∈ X, n ∈ N, k and hn is Σγnk+1(F)–measurable, where γnk = max{γn(k−1), αnk}, n ∈ N. Having this done, we set k hk(x) = hk(x), x ∈ X, k ∈ N.

Then the mappings hk are Σγkk+1(F)–measurable, where γkk < α, and hk → f.

According to the inductive assumption, hk ∈ Φγkk , k ∈ N. Hence f ∈ Φα as required.

3. Baire, Borel and resolvable sets If X is a topological space, we write G(X) for the sublattice of all open subsets of X. A subset A of X is called a zero set if A = f −1({0}) for a continuous real-valued function f on X. It is clear that such a function f can be chosen with values in [0, 1]. A cozero set is the complement of a zero set. It is easy to check that zero sets are preserved by finite unions and countable intersections. Hence cozero sets are preserved by finite intersections and countable unions. (Zero and cozero sets are termed functionally closed and functionally open sets in [4, p. 42]). We recall that Borel sets is the σ–algebra generated by the family of all open subset of X and Baire sets is the σ–algebra generated by the family of all cozero sets in X. We recall that a subset A of a topological space X is Fσ, if A can be written as a countable union of closed sets. The complement of an Fσ set is called a Gδ set. The space X is called scattered if its each subset has an isolated point, that is, for each A ⊂ X there exists x ∈ A and an open set U such that A ∩ U = {x}. The space X is σ–scattered, if X can be written as a countable union of scattered subspaces. Lemma 3.1. Let F be a sublattice of sets in a set X. The the smallest algebra generated by F consists of all finite unions of differences of sets from F. Proof. Let B denote the family of all finite unions of differences from F and let A be the algebra generated by F. Obviously, B⊂A. n Let B = i=1(Fi \ Hi), where the sets Fi, Hi are contained in F. Then S n X \ B = (H ∪ (X \ F )) = H \ F . i i  i i i=1 i I \ I⊂{[1,...,n} \∈ i∈{1,...,n[ }\I   Hence B is stable with respect to complements and, clearly, to finite unions. Thus B is an algebra and A ⊂ B.

Deﬁnition 3.2. We consider the following families of subsets of X. (a) The algebra Bas(X) generated by zero sets. By Lemma 3.1, n Bas(X) = { (Fi \ Hi) : Fi, Hi are zero sets in X, n ∈ N}. i=1 [ BOREL SETS AND FUNCTIONS IN TOPOLOGICAL SPACES 7

(b) The algebra Bos(X) generated by closed subsets of X. As above,

n Bos(X) = { (Fi \ Hi) : Fi, Hi are closed in X, n ∈ N}. i=1 [ (c) The algebra Hs(X) of all H–sets (or resolvable sets). H–sets are defined in [31, §12, II], where their basic properties are described (see also [29, p. 218]). Let us recall some equivalent definitions. A subset A of a topological space X is an H–set if for any nonempty B ⊂ X there is a nonempty relatively open U ⊂ B such that either U ⊂ A or U ∩A = ∅. It is clear that H–sets form an algebra containing all open sets. Further, A is an H–set in X if and only if A is the union of a scattered family of sets of the form F ∩ G with F closed and G open. (We recall that a family U of subsets of a topological space is scattered if it is disjoint and for each nonempty V⊂U there is some V ∈ V relatively open in V. Thus it follows that a topological space X is scattered if {{x} : x ∈ X} is a scattered family.) S For each algebra of sets listed in (a)–(c) we consider the classes of sets defined in Definition 2.1.

(d) If we start the Borel hierarchy as defined in Definition 2.1 from the sublattice G(X) of all open subsets of X, for metrizable spaces we get the standard 0 Borel hierarchy as defined in [28, Section 11.B]. We write Σα(G(X)) and 0 Πα(G(X)) for the families obtained by this procedure. We show below its relation to the families defined in (a)–(c). We just mention that a set A 0 belongs to Σ2(G(X)) if and only if A is of type Fσ. Remark 3.3. In general, Hs(X) may contain a non-Borel set, and thus Hs(X) may be a strictly larger family than the system of all Borel sets in X. An easy example is provided by a suitable scattered compact space X. Namely, in this case any subset of X is an H–set, since {{x} : x ∈ X} is a scattered family consisting of closed sets. If X = [0, ω1] with the order topology and A ⊂ [0, ω1) is a stationary subset, so that [0, ω1) \ A is also stationary (see [27, Lemma 7.6]), then A is a resolvable non-Borel set (see [42, Lemma 1], [16, p. 296] or [22, Example 4.4]).

Proposition 3.4. Let X be a Tychonoﬀ space. Then the following assertions hold:

(a) Σα(Bas(X)) ⊂ Σα(Bos(X)) ⊂ Σα(Hs(X)), α ∈ (0, ω1),

(b) α<ω1 Σα(Bas(X)) is the σ–algebra of all Baire sets in X and the family Σα(Bos(X)) is the σ–algebra of all Borel sets in X, Sα<ω1 (c) if A is a subset of a normal space, then A ∈ ∆ (Bas(X)) if and only if A S 2 is both Fσ and Gδ, (d) if X is metrizable, then (d1) Σα(Bas(X)) = Σα(Bos(X)), α ∈ (0, ω1), 0 (d2) Σα(Hs(X)) = Σα(Bos(X)) = Σα(G(X)), α ∈ (1, ω1); (d3) if X is completely metrizable, then Hs(X) = ∆2(Bos(X)). Proof. Assertion (a) follows from the fact that Bas(X) ⊂ Bos(X) ⊂ Hs(X) and (b) follows from Proposition 2.3(d). For the proof of (c), any set in ∆2(Bas(X)) is both ∞ ∞ Fσ and Gδ in any topological space. If X is normal and A = n=1 Fn = n=1 Gn, where Fn ⊂ X closed, Gn ⊂ X open, n ∈ N, then there exist continuous functions S T 8 JIRˇ´I SPURNY´ fn : X → [0, 1] so that

1, on Fn, fn = n ∈ N. (0, on X \ Gn,

Then fn → χA, and hence

∞ ∞ 1 ∞ ∞ 1 A = {x ∈ X : f (x) ≥ } = {x ∈ X : f (x) > } n 2 n 2 k n k k n k [=1 \= \=1 [= is in ∆2(Bas(X)). Since (d1) follows from the fact that any closed subset of a metrizable space is a zero set, we proceed to the proof of (d2). We show that any set A ∈ Hs(X) is both Fσ and Gδ in case X is metrizable. Indeed, if A ∈ Hs(X), A is a scattered union of Fσ sets, and thus is also of type Fσ by Montgomery’s lemma [39, Lemma 16.2]. Thus Σ2(Hs(X)) ⊂ Σ2(Bos(X)), which along with (a) gives the first equality. For the second equality we remark that any Fσ set is in Σ2(Bos(X)) and any set in Σ2(Bos(X)) is of type Fσ. Hence the assertion follows by transfinite induction. Let X be completely metrizable and A ∈ Hs(X). According to the abovemen- tioned Montgomery’s lemma [39, Lemma 16.2], A is both Fσ and Gδ, and hence in ∆2(Bos(X)). Conversely, if A ∈ ∆2(Bos(X)), then its characteristic function χA is a function of the first Baire class by [28, Theorem 24.3]. By [28, Theorem 24.14], χA has a point of continuity when restricted to any closed set F ⊂ X. But this means nothing else than that A ∈ Hs(X). This concludes the proof.

Remark 3.5. We remark that other definitions of Borel classes can be found in [17, Section 3], [40, Definition 1.1] and [23, p. 10]. We recall these definitions and describe the relations of the resulting Borel families with our classes. The Borel families Bα, Cα in a topological space X are defined in [17, Section 3, p. 25] as follows:

B0 = {F ∩ G : F closed in X, G open in X} and, if α = β + 1, then

(B ) , α is odd, B = β σ α B (( β)δ, α is even; B B if α is limit, then α = β<α β. Finally,

CαS= {X \ B : B ∈ Bα}, α ∈ [0, ω1). It follows from Deﬁnition 3.2 that B , n is odd, Σ (Bos(X)) = n n ∈ . n+1 C N ( n, n is even,

For a limit ordinal λ ∈ [ω0, ω1) we get

C , n is odd, Σ (Bos(X)) = λ+n+1 n ∈ ∪ {0}. λ+n B N ( λ+n+1, n is even, BOREL SETS AND FUNCTIONS IN TOPOLOGICAL SPACES 9

The classes Aα, Mα, α ∈ [0, ω1), are deﬁned in [40, Deﬁnition 1.1] as follows: A0 consists of all open sets, M0 consists of all closed sets and, for α ∈ (0, ω1),

∞

Aα = { (An ∩ Bn) : An ∈Aαn ,Bn ∈ Mαn , αn < α}, and n=1 [∞

Mα = { (An ∪ Bn) : An ∈Aαn ,Bn ∈ Mαn , αn < α}. n=1 \ Our deﬁnition of Borel classes Σα(Bos(X)) coincides with the Borel classes Aα(X) deﬁned on [23, p. 10] except for the fact that Σn+1(Bos(X)) = An(X) for n ∈ N. Thus it follows from [23, Remark, p. 11] that

Σα+1(Bos(X)), α ∈ (0, ω0), Aα = (Σα(Bos(X)), α ∈ [ω0, ω1). Theorem 3.6. Let A be a Baire subset of a compact space X.

(a) If α ∈ (1, ω1), then the following assertions are equivalent: (i) A ∈ Σα(Bas(X)), (ii) A ∈ Σα(Bos(X)), (iii) A ∈ Σα(Hs(X)). (b) A ∈ Bas(X) if and only if A ∈ Bos(X). (c) A ∈ Hs(X) if and only if A ∈ ∆2(Bas(X)). Proof. To prove (a) we need to verify the only nontrivial implication (iii) =⇒ (i). To this end, let A be a Baire subset of a compact space X and α ∈ (1, ω1). Assume that A ∈ Σα(Hs(X)). Since A is Baire, we can ﬁnd a countable family {fn : n ∈ N} of real–valued continuous functions on X such that A belongs to the σ–algebra −1 N generated by {fn ({0}) : n ∈ N}. We deﬁne a mapping ϕ : X → [0, 1] as

ϕ(x) = {fn(x)}n∈N, x ∈ X. Then ϕ is a continuous mapping of X onto a compact metrizable space Y = ϕ(X). Further, ϕ−1(ϕ(A)) = A, and so ϕ is a perfect mapping of A onto ϕ(A) (that is, ϕ is continuous, maps closed sets to closed sets and the ﬁber ϕ−1(y) is compact for each y ∈ ϕ(A), see [4, Section 3.7] for more information on perfect mappings). According to [23, Theorem 4], ϕ(A) ∈ Σα(Hs(Y )). By Proposition 3.4(d), −1 ϕ(A) ∈ Σα(Bas(Y )). Since A = ϕ (ϕ(A)), A ∈ Σα(Bas(X)). This concludes the proof of (a). The proof of (b) and (c) can be done by the same method, we use Proposi- tion 3.4(d1) and (d3) instead.

There are many descriptive properties that are “absolute” in a sense that a Tychonoff space X has a property (P) in its Cech–Stoneˇ compactification if and only if X has (P) in every Y containing X. As a classical example serves Cechˇ completeness, that is, X is a Gδ subset of its Cech–Stoneˇ compactification if and only if X is a Gδ subset of every Tychonoff space Y containing X (see [4, Section 3.9]). For more involved examples of this phenomenon see [24, Theorem 8], [43, Theorems 5.8.6, 5.8.7 and 5.8.8], [19, Theorem 3], [20, Theorem 2], [16, Theorem 5.3 and Theorem 6.14(d)] or [3, Sections 5 and 6]. The same can be proved e.g. for 10 JIRˇÍ SPURNY´

• a space obtainable as a result of the Souslin operation applied to closed sets in its Cech–Stoneˇ compactification (so–called K–analytic spaces, see [43, Theorem 5.8.8]), • a space obtainable as a result of the Souslin operation applied to Borel sets in its Cech–Stoneˇ compactification (so–called Cechˇ analytic spaces, see [11] and [23, Theorem 5]), • a space obtainable as a result of the Souslin operation applied to resolvable sets in its Cech–Stoneˇ compactification (such spaces are called scattered–K– analytic spaces in [16, Theorem 5.3 and Theorem 6.14(d)] and [23, Theorem 5], almost K–descriptive in [19, Theorem 3] and cover–analytic in [38, §2]), • a space of some Borel class in its Cech–Stoneˇ compactification (see [23, Corollary 14]), • a space of some resolvable class in its Cech–Stoneˇ compactification (see [23, Theorem 5]). The following result on Baire classes is in the same spirit as the assertions above.

Theorem 3.7. For a Tychonoﬀ space X and α ∈ (0, ω1) the following assertions are equivalent:

(i) X ∈ Σα(Bas(Y )) in every Tychonoff space Y containing X as a dense subset, (ii) X ∈ Σα(Bas(Y )) in every compactification Y of X, (iii) X ∈ Σα(Bas(Y )) in some compactification Y of X. Proof. The only nontrivial implication is (iii) =⇒ (i). We assume that X ∈ Σα(Bas(Y )) in some compactification Y of X. Let βX be the Cech–Stoneˇ compactification of X and ϕ : βX → Y be the continuous mapping so that ϕ is identity on X (see [4, Corollary 3.6.1]). Then ϕ(βX \ X) = Y \ X (see [4, Theorem 3.7.16]), and thus X ∈ Σα(Bas(βX)). Let Z be any Tychonoff space containing X as a dense subset and let βZ be its Cech–Stoneˇ compactification. Then βZ is a compactification of X, and so there exists a continuous mapping ψ : βX → βZ such that ψ equals identity on X and ψ(βX \ X) = βZ \ X. Since both X and βX \ X are K–analytic in βX (see [43, Theorem 2.5.1]), X and βZ \ X are K–analytic in βZ. By the separation theorem [7, Theorem 1] (or [43, Theorem 5.1.6(c)]), X is a Baire subset of βZ. By Proposition 3.4(a), X ∈ Σα(Hs(βX)). According to [23, Theorem 4], X ∈ Σα(Hs(βZ)). It follows from Theorem 3.6(a) that X ∈ Σα(Bas(βZ)). Hence X ∈ Σα(Bas(Z)), which concludes the proof.

Remark 3.8. We remark that Theorem 3.7 also follows from the results by J. Saint– Raymond ([45, Th´eor`eme 5]) and J.E. Jayne and C.A. Rogers (see [43, Theorem 5.9.13]).

4. Baire, Borel and resolvable order Deﬁnition 4.1. If F is a family of sets in a set X, the order of F (denoted as ord(F)) is the least α ∈ (0, ω1) such that Σα(F) = Σα+1(F) if such α exists, otherwise the order is ω1 (see [34, p. 430]). If X is a topological space, we call ord(Bas(X)), ord(Bos(X)) and ord(Hs(X)) the Baire, Borel and resolvable order of X, respectively. BOREL SETS AND FUNCTIONS IN TOPOLOGICAL SPACES 11

If X is an inﬁnite compact space, it is well known that ord(Bas(X)) is either 2 or ω1, depending on the fact whether X is scattered or not (see [2], [36, Theorem 3.4], [43, Theorem 6.1.2] or Theorem 4.2 below). The question of the possible values of the Borel order of a compact space X is asked in [34, Question, p. 440] and [35, Problem, p. 395]. The following theorem solves one part of this problem. Theorem 4.2. For a Tychonoﬀ space X the following assertions hold. (a) If X is a K–analytic σ–scattered space, then ord(Bas(X)) = 2 and ord(Hs(X)) = 2. (b) If X contains a compact set, that can be mapped continuously onto a compact metrizable perfect space, then

ord(Bas(X)) = ord(Bos(X)) = ord(Hs(X)) = ω1. Proof. For the proof of (a), let X be a K–analytic σ–scattered space. Then ∞ X = n=1 Xn for some scattered subspaces Xn, n ∈ N. Since any subset of a scattered space is a resolvable set, any subset of X belongs to Σ2(Hs(X)), and thus ord(Hs(SX)) = 2. Further, let A ⊂ X be a Baire set. Then it is easy to ﬁnd countably many real–valued continuous functions fn, n ∈ N, on X such that the mapping N ϕ : X → R ,

x 7→ {fn(x)}n∈N satisfies ϕ−1(ϕ(A)) = A (see [43, proof of Theorem 5.9.13]). Then Y = ϕ(X) is a metrizable K–analytic space that has no compact perfect subsets. (Indeed, X does not contain a perfect compact subset by [30, Theorem 3.1] and thus the assertion follows from [43, Theorem 5.4.2].) Hence Y is an analytic set with no compact perfect subsets and thus Y is countable by [28, Theorem 29.1]. −1 Thus ϕ(A) ∈ Σ2(Bas(Y )), and hence A = ϕ (ϕ(A)) ∈ Σ2(Bas(X)). For the proof of (b), we assume that there is a compact set K ⊂ X that is continuously mapped onto a compact metrizable perfect space L by a mapping ϕ. 0 We fix α ∈ (1, ω1) and choose a Borel set B ⊂ L such that B∈ / Σα(G(L)) (see [28, Theorem 22.4]). Then A = ϕ−1(B) is a Baire subset of K that does not belong to Σα(Bas(K)). (Indeed, if it were true, then 0 B = ϕ(A) ∈ Σα(Hs(L)) = Σα(G(L)) by [23, Theorem 4] and Proposition 3.4(d3), which is not the case.) By Theo- rem 3.6(a), A∈ / Σα(Hs(K)) ∪ Σα(Bos(K)), and hence A∈ / Σα(Hs(X)) ∪ Σα(Bos(X)). Finally, it is easy to observe that any Baire subset of K is a trace of a Baire subset of X (use Tietze’s theorem for the set K considered as a closed subset of the Cech–ˇ Stone compactification of X). Hence there exists a Baire set C ⊂ X such that C∈ / Σα(Bas(X)). This concludes the proof. If we combine [38, Theorem 5.4] with Theorem 4.2(b), we get the following corollary. (We recall that almost Cech–analyticˇ spaces were defined in [38, §4] and that this class of spaces contains scattered–K–analytic spaces.) 12 JIRˇÍ SPURNY´

Corollary 4.3. If X is an almost Cech–analyticˇ spaces, that is not σ–scattered, then ord(Bas(X)) = ord(Bos(X)) = ord(Hs(X)) = ω1. If we consider the compact scattered space X = [0,κ] for some ordinal κ (en- dowed with the order topology), then any Borel subset of X belongs to Σ2(Bos(X)) (see [42, Lemma 1] and [35, Theorem 4]). Thus ord(Bos(X)) = 2. Nevertheless, it seems to be an open problem whether the same conclusion can be obtained for an arbitrary scattered space. Question 4.4. Let X be a compact scattered space. Is it true that ord(Bos(X)) = 2?

5. Baire and Baire–one mappings Deﬁnition 5.1. We consider the following classes of mappings between topological space X and Y .

(a) Let Baf1(X,Y ) be the family of all Σ2(Bas(X))–measurable mappings from X to Y and for α ∈ (1, ω1), let Bafα(X,Y ) = (Baf1(X,Y ))α (see Deﬁni-

tion 2.4 and Remark 2.5). We call the elements of α<ω1 Bafα(X,Y ) the Baire measurable mappings. S (b) Let Bof1(X,Y ) be the family of all Σ2(Bos(X))–measurable mappings from X to Y and for α ∈ (1, ω1), as above we set Bofα(X,Y ) = (Bof1(X,Y ))α.

We call the elements of α<ω1 Bofα(X,Y ) the Borel measurable mappings. (c) Let Hf (X,Y ) be the family of all Σ (Hs(X))–measurable mappings from 1 S 2 X to Y and for α ∈ (1, ω1), as above we set Hfα(X,Y ) = (Hf1(X,Y ))α. We

call the elements of α<ω1 Hfα(X,Y ) the resolvably measurable mappings. The following theorem justifiesS the term “measurability” in Definition 5.1. Theorem 5.2. Let f be a mapping from a Tychonoff space X to a separable metrizable space Y and α ∈ (0, ω1). Then the following assertions hold:

(a) f ∈ Bafα(X,Y ) if and only if f is Σα+1(Bas(X))–measurable. (b) f ∈ Bofα(X,Y ) if and only if f is Σα+1(Bos(X))–measurable. (c) f ∈ Hfα(X,Y ) if and only if f is Σα+1(Hs(X))–measurable. Proof. It follows from Theorem 2.7.

Deﬁnition 5.3. Let α ∈ (0, ω1). A mapping f : X → Y between topological spaces X and Y is said to be of Baire class α if f ∈ (C(X,Y ))α, where C(X,Y ) denotes the set of all continuous mappings from X to Y . We write Cα(X,Y ) for the family of all mappings of Baire class α. Lemma 5.4. Let X be a K–analytic space, Y a metric space and f : X → Y be a mapping such that f −1(U) is a Baire subset of X for every open U ⊂ Y . Then f(X) is separable. Proof. Let f : X → Y be as in the premise and let G be a disjoint family of open sets in Y . Then F = {f −1(G) : G ∈ G} is a disjoint family in X such that F ′ is a Baire set in X for each F ′ ⊂F. By virtue of [9, Lemma 1] (see also [10, Theorem 3E]), F is countable. S If B is a σ–discrete base in Y (see [4, Theorem 4.4.1]), it follows that f(X) intersects only countably many members of B. Hence f(X) is a separable subspace of Y . BOREL SETS AND FUNCTIONS IN TOPOLOGICAL SPACES 13

Theorem 5.5. Let X be a compact space, Y a metric space and f : X → Y be a mapping such that f −1(U) is a Baire subset of X for every open U ⊂ Y . Let α ∈ (0, ω1). Then the following assertions are equivalent:

(i) f ∈ Bafα(X,Y ), (ii) f ∈ Bofα(X,Y ), (iii) f ∈ Hfα(X,Y ). Proof. Obviously, (i) =⇒ (i) =⇒ (iii). For the proof of (iii) =⇒ (i), let f ∈ Hfα(X,Y ) for some α ∈ (0, ω1). By virtue of Lemma 5.4, the range f(X) is −1 separable. By Theorems 5.2 and 3.6, f (U) ∈ Σα+1(Bas(X)) for any open set U ⊂ Y . Hence f ∈ Bafα(X,Y ) by Theorem 5.2.

The following theorem is a variant of the classical characterization of mappings of Baire class α via their measurability (see e.g. [28, Theorem 24.3] or [5, Theorem 3]). Corollary 5.6. Let X be a compact space, Y be an arcwise connected locally arcwise connected metric space Y and f : X → Y be a function such that f −1(U) is a Baire subset of X for every open U ⊂ Y . Let α ∈ (0, ω1). Then the following assertions are equivalent:

(i) f ∈ Bafα(X,Y ), (ii) f ∈ Bofα(X,Y ), (iii) f ∈ Hfα(X,Y ), (iv) f ∈Cα(X,Y ). Proof. According to Theorem 5.5, assertions (i)–(iii) are equivalent. If α = 1 and f : X → Y is a Σ2(Bas(X))–measurable mapping from a compact space X to an arcwise connected locally arcwise connected metric space Y , Lemma 5.4 allows to use [49, Theorem 3.7] to conclude that f ∈C1(X,Y ). Thus f ∈ Bafα(X,Y ) if and only if f ∈Cα(X,Y ) for any α ∈ [1, ω1).

Remark 5.7. The most difficult part of the proof of Corollary 5.6 is to show that C1(X,Y ) equals the space of Σ2(Bas(X))–measurable mappings. There is a long series of papers devoted to the question under what conditions a function f : X → Y −1 is of Baire class 1 if and only if f (U) is Fσ for each U ⊂ Y open. This question has an affirmative answer in any of the following situations: • X is an interval in R and Y = R (see [1]), • X is metric, Y = R (see [32]), N N • X is metric, Y = [0, 1] , n ∈ N, or Y = [0, 1] (see [31, §IX]), • X is metric, Y is a separable convex subset of a Banach space (see [44, Lemma 3]), • X is a complete metric space and Y is a Banach space (see [47, Theorem 4]), • X is a normal topological space, Y = R (see [15] or [33, Exercise 3.A.1]), If f : X → Y is σ–discrete (see [15, §3], [26, Section 2.2] or [49, p. 2] for the definition and basic properties), then f is of Baire class 1 if and only if f −1(U) is Fσ for each U ⊂ Y open in any of the following situation: • X is a perfectly normal paracompact space, Y is a Banach space (see [25, Corollary 7]), 14 JIRˇÍ SPURNY´

• X is collectionwise normal and Y is a closed convex subset of a Banach space (see [15]), • X is metric, Y is a complete connected and locally connected metric space (see [5, Theorem 2]), • X is normal and Y is arcwise connected and locally arcwise connected (see [49, Theorem 3.7]). Theorem 5.8. Let f : X → Y be a Baire function from a compact space X to a metric space (Y, ρ). Then the following assertions are equivalent:

(i) f ∈ Baf1(X,Y ), (ii) f|F has a point of continuity for every closed F ∈ X, (f has the point continuity property), (iii) for each ε > 0 and F ⊂ X there exists a relatively open set U ⊂ F such that diamρ f(U) < ε (f is fragmented), (iv) the set {x ∈ F : f|F discontinuous at x} is meager in F for each F ⊂ X, (v) f −1(G) has the restricted Baire property (that is, f −1(G)∩F has the Baire property in F for each F ⊂ X) for each G ⊂ Y open. If Y is arcwise connected and locally arcwise connected, then (i) is also equivalent to

(vi) f ∈C1(X,Y ). Proof. If f : X → Y is a Baire function from a compact space X to a metric space Y , its range f(X) is separable by Lemma 5.4. By Theorem 5.5, f ∈ Baf1(X,Y ) if and only if f ∈ Hf1(X,Y ), and this is the case if and only if f is Σ2(Hs(X))– measurable. Thus it follows from [29, Theorem 2.3] that all conditions (i)–(v) are equivalent. If Y is moreover arcwise connected and locally arcwise connected, Corollary 5.6 yields (i)⇐⇒(vi), ﬁnishing thus the proof.

6. Examples and questions In this section, we present several examples showing that Baire and Borel classes might differ quite significantly, when we do not work within the frame of compact spaces. The following example shows that the difference of Baire and Borel classes might be arbitrarily high even in normal spaces. It is taken from [50, Section 1], eventhough this space was already used by E. Michael in [37].

Example 6.1. For every α ∈ (2, ω1), there exists a Baire subset A of a normal space X such that A is of additive Borel class 1 and A is not of additive Baire class α.

Proof. We briefly recall the construction of S. Willard. We fix α ∈ (2, ω1). If ǫ denotes the Euclidean topology of the real line R, let A ⊂ R be a set in Σα(Bas(R, ǫ))\ Πα(Bas(R, ǫ)) (see [28, Theorem 22.4]). Let τ be the topology on R such that its open sets are of the form U ∪ V , where U is ǫ–open and V is any subset of R \ A. Then (R, ǫ) is a normal space (see [37, p. 375]). Since τ is finer that ǫ, A is a Baire subset of (R,τ). Further, A is τ–closed. We claim that A∈ / Σα(Bas(R,τ)). To see this, we notice that for any τ–open V ⊂ R there exists an ǫ–open set W ⊂ R such that V ∩ A = W ∩ A and W ⊂ V . Thus, given a τ–continuous function f on R, for every η > 0 there exists an ǫ–open set Uη ⊃ A such that for each x ∈ Uη there exists an open interval I satisfying BOREL SETS AND FUNCTIONS IN TOPOLOGICAL SPACES 15 x ∈ I ⊂ Uη and diam f(I) <η. Hence it follows that for f there exists a Gδ set G in (R, ǫ) such that A ⊂ G and f|G is ǫ–continuous. Thus A ∈ Σβ(Bas(R,τ)) if and only if A ∈ Σβ(Bas(R, ǫ)), β ∈ (2, ω1). This concludes the proof. If A is a subset of a normal space X such that both A ⊂ X and X \ A are of type Fσ, then both A and X \ A are countable unions of zero sets. This follows by Tietze’s theorem (see Proposition 3.4(c)). Thus the following example might be of some interest because it shows that an analogous assertion is false for higher classes. Example 6.2. There exists a set A in a normal space such that both A and its complement is of type Fσδ and A is not a Baire set (we recall that A is of type Fσδ ∞ ∞ if A = n=1 m=1 Fnm for some closed sets Fnm, n, m ∈ N). Proof. TWe considerS the following combination of the Alexandroff double of the unit interval (see [4, Example 3.1.26]) and the discretization procedure from [37]. Let A ⊂ [0, 1] be a set chosen in such a way that

A ∈ Σ3(Bas(([0, 1], ǫ))) \ Π3(Bas([0, 1], ǫ))). We notice that A is uncountable. Let B = [0, 1] \ A and X = [0, 1] × {0, 1} considered with the following topology τ: Points of [0, 1] × {1} and B × {0} are isolated and any point (x, 0) ∈ A × {0} has a base of neighborhood consisting of sets X ∩ (x − ε, x + ε) × {0} ∪ ((x − ε, x + ε) × {1}) \ K , ε > 0, K finite. Then (X,τ) is obviously a regular space that is moreover normal. Indeed, it is enough to verify that (X,τ) is paracompact (see [4, Theorem 5.1.5]). If U is a τ– open cover of X, we first select a countable τ–open family V such that {(A × {0}) ∩ V : V ∈ V} refines {(A × {0}) ∩ U : U ∈ U} (this is possible because (A × {0},τ) is a separable metrizable space). Since X \ (A × {0}) is a discrete space, V ∪ {{x} : x ∈ X \ V)} is a σ–discrete τ–open refinement of U and hence[ (X,τ) is paracompact by [4, Theorem 5.1.12]. Claim 6.2.1. Let f : (X,τ) → R be continuous and δ > 0. Then there exists a ǫ–open set G ⊂ [0, 1] and a countable set S ⊂ G such that A ⊂ G and |f((x, 0)) − f((x, 1))| ≤ δ for each x ∈ G \ S. Proof of Claim 6.2.1. Let f : (X,τ) → R be continuous and δ > 0. It follows from the definition of τ that for any x ∈ A there exists an interval Ux ⊂ [0, 1] so that |f((y, 0)) − f((y, 1))| < δ for each y ∈ A ∩ (Ux \ {x}). We choose countably many ∞ ∞ points xn ∈ A, n ∈ N, so that A ⊂ n=1 Uxn . Then the sets U = n=1 Uxn and S = {xn : n ∈ N} are the required ones. S S Claim 6.2.2. Let fn : (X,τ) → R, n ∈ N, be continuous functions. Then there exist a Gδ set G ⊂ ([0, 1], ǫ) and a countable set S ⊂ G such that A ⊂ G and fn((x, 0)) = fn((x, 1)) for each x ∈ G \ S and n ∈ N. 16 JIRˇÍ SPURNY´

Proof of Claim 6.2.2. Let fn : (X,τ) → R, n ∈ N, be continuous functions. Using Claim 6.2.1 we ﬁnd Gδ sets Gnm ⊂ [0, 1] and countable sets Snm ⊂ Gnm, n, m ∈ N, such that 1 |fn((x, 0)) − fn((x, 1))| ≤ , x ∈ Gnm \ Snm and Anm ⊂ Gnm, n,m ∈ N. m Then the sets ∞ ∞ G = Gnm and S = Snm n,m=1 n,m=1 \ [ fullﬁl our requirements.

Claim 6.2.3. The set B × {0} is a τ–open set that is of type Fσδ in (X,τ).

Proof of Claim 6.2.3. Obviously, B × {0} is open. Further, let Fnm, n, m ∈ N, be ∞ ∞ ǫ–closed sets in [0, 1] such that B = n=1 m=1 Fnm. Then ∞ ∞ T S B × {0} = ([0, 1] × {0}) ∩ (Fnm × {0, 1}), n=1 m=1 \ [ and B × {0} is of type Fσδ in (X,τ). It follows from the preceding claim that both B × {0} and X \ (B × {0}) are of type Fσδ in (X,τ). To finish the proof we have to verify that B × {0} is not a Baire subset of (X,τ). Assume that there exists a countable family {fn : n ∈ N} of τ–continuous functions on X such that χB×{0} is contained in the closure of {fn : n ∈ N} with respect to the topology of pointwise convergence. Let G and S be the sets provided by Claim 6.2.2. If G \ A were contained in S, A would be in Π3(Bas(([0, 1], ǫ))), a contradiction. Hence there exists a point x ∈ G\A such that fn((x, 0)) = fn((x, 1)), n ∈ N. But this is impossible as χB×{0} is in the pointwise closure of {fn : n ∈ N}. This finishes the proof. The following example shows that there there exist sets in compact spaces that are “very far” from being Baire in spite of the fact that they are of very low Borel class and K–analytic. The method of the construction is a well know Lindelöf modification that can be found e.g. in [43, Example 5.2.3] or [12, p. 2]. Example 6.3. There exists a K–analytic subset of a compact space of additive Borel class 1 that is not Baire. N Proof. Let X be the space constructed in [43, Example 5.2.3], that is, X = N ∪{p}, N N where p is a point not contained in N with the following topology τ: Points of N are isolated and a base of neighborhoods of a point p is formed by sets N N N {p} ∪ (N \ F ), F ⊂ N is closed and discrete in the usual topology of N .

Then (X,τ) is a K–analytic space that is of type Kσδ in its Cech–Stoneˇ compacti- ∞ ∞ ﬁcation βX, that is, X = n=1 k=1 Knk, where the sets Knk are compact in βX (see [43, Example 5.2.3]). On the other hand, X is not a Baire subset of βX (see again [43, Example 5.2.3] forT theS proof of this fact). We claim that X is a set in βX of the ﬁrst Borel class. Indeed, X \ {p} is an open set in βX, and hence X is the union of an open and closed set. This concludes the proof. BOREL SETS AND FUNCTIONS IN TOPOLOGICAL SPACES 17

It is a natural question whether the results of Theorem 3.6 can be proved for more general spaces then just compact ones. A natural candidate is the class of K–analytic spaces. In particular, we do not know the answer to the following question. Question 6.4. Let A be a subset of a K–analytic space X such that A is a Baire subset of X and A ∈ Σ1(Bos(X)). Does A have to be in Σ1(Bas(X))? The situation would be clear if the complexity of a Baire set in a K–analytic space could be determined by its behaviour on compact sets. An aﬃrmative answer to the following question would yield the results of Theorem 3.6 for K–analytic spaces.

Question 6.5. Let α ∈ (1, ω1) and let A be a Baire subset of a K–analytic space X such that A∈ / Σα(Bas(X)). Does there exist a compact set K ⊂ X such that A ∩ K∈ / Σα(Bas(K))? Acknowledgement. The author would like to express his gratitude to his col- leagues at the Faculty of Mathematics and Physics of Charles University who helped him substantially during the work on the paper. In particular, he would like to thank Professors P. Holick´y, M. Huˇsek, O. Kalenda, P. Simon and M. Zelen´y for fruitful discussions and remarks on the subject.

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Charles University, Faculty of Mathematics and Physics, Department of Mathemat- ical Analysis, Sokolovska´ 83, 186 75 Praha 8, Czech Republic