Countable open and closed functions

arXiv:1102.1954v3 [math.GN] 24 Nov 2011 PRESERVATION OF THE BOREL CLASS UNDER OPEN-LC FUNCTIONS ALEXEY OSTROVSKY Abstract. Let X be a Borel subset of the Cantor set C of additive or multiplicative class α, and f : X →Y be a continuous function onto Y ⊂C with compact preimages of points. If the image f(U) of every clopen set U is the intersection of an open and a closed set, then Y is a Borel set of the same class α. This result generalizes similar results for open and closed functions. 1. Introduction Let X be a Borel subset of the Cantor set C of additive or multiplicative class α, and f : X →Y be a continuous function onto Y ⊂C with compact preimages of points. It is well known that if the image f(U) of every clopen set U is an open subset of Y , then Y is a Borel set of the same class α [9], [8], [1], [7]. Analogously, if the image f(U) of every clopen set U is a closed subset of Y , then Y is a Borel set of the same class. The aim of this note is to prove (Theorem 2) that if the image f(U) of every clopen set U is an intersection of an open and a closed set, then Y is a Borel set of the same class. This fact is related to the following problem [6, Problem 3.6.]: Find a class of open-Borel, continuous functions that are close as possible to open and closed functions and have compact preimages of points and preserve absolute Borel class. 2. Related materials and basic definitions All spaces in this paper are assumed to be metrizable and separable. Recall that a subset of a topological space is an LC-set or a locally closed set if it is the intersection of an open and a closed set. Given an arbitrary (not necessarily continuous) function f we say that it is -open (resp. closed) if f takes open (resp. closed) sets into open (resp. closed) sets; -open(resp. clopen)-LC if f takes open (resp. clopen) sets into LC-sets. The following assumptions will be needed throughout the paper. 2000 Mathematics Subject Classification. Primary 54C10; Secondary 54H05, 54E40, 03E15. Key words and phrases. Borel sets, locally closed sets, clopen sets, open and closed functions, Borel isomorphism. 1 2 ALEXEY OSTROVSKY We will denote by S1(y) a sequence with its limit point: S1(y) = {y} ∪{yi : yi −→y} . It is easy to check that a function f is closed ⇔for every S1(y), every sequence xi ∈f −1(yi) ( yi ̸= yj for i ̸= j) has a limit point in f −1(y); Indeed, if f is closed and, for some S1(y), there is no limit point in f −1(y) for xi ∈f −1(yi), then the image f(T) of the closed set T = clX{xi} is not closed in Y . Conversely, if there is a closed T ⊂X for which f(T) is not closed in Y , then there is S1(y) such that y ̸∈f(T) and yi ∈f(T). Hence, since T is closed, every sequence of points xi ∈f −1(yi) ∩T has no limit point in f −1(y). Analogously, it is easy to check that a function f is open ⇔for every S1(y) and every open ball O(x), x ∈f −1(y), there are only finitely many yi such that f −1(yi) ∩O(x) = ∅. 3. Structure of clopen-LC functions in the Cantor set C Let us first prove the following theorem. Theorem 1. Let f : X →Y be a clopen-LC function from a subset X of the Cantor set C onto Y and the inverse image of every point y be com- pact. Then Y can be covered by countably many subsets Yn such that the restrictions f|f −1(Yn) are open functions (n = 1, 2, ...) and the restriction f|f −1(Y0) is a closed function. Proof. Denote A. Xn = S{f −1(y) : there is S1(y) ⊂Y and ˜xy ∈C such that there are xk ∈f −1(yk) with xk −→˜xy and dist(˜xy, f −1(y)) > 1/n. Lemma 1. The restriction f|Xn is an open function onto Yn = f(Xn). Indeed, suppose the opposite. Then B. for some y ∈f(Xn) and d > 0, there is S1(y) ⊂f(Xn) and x ∈f −1(y) such that yk −→y and dist(x, f −1(yk)) > d. Let us consider a countable compact set S2(y) obtained by replacing (see item B) the isolated points yk of S1(y) by S1(yk) ⊂Y with isolated points ykj −→yk selected according to item A. Since X lies in C there is a limit point ˜x ∈C for ˜xyk. The proof falls naturally into two parts. (1) If ˜x ̸∈X, then we can take a clopen (in C) ball Oδ1(˜x), δ1 < 1/n, and a clopen (in C) ball Oδ2(x), where δ2 < d, according to B. It is clear that D = Oδ1(˜x) ∪Oδ2(x) is a clopen set in C and hence S2(y) ∩f(D) is the intersection of a closed set F and an open set U in S2(y). We can suppose that S2(y) ∩f(D) contains y, ykj (j, k = 1, 2, ...), and obviously yk ̸∈f(D). PRESERVATION OF THE BOREL CLASS UNDER OPEN-LC FUNCTIONS 3 Since the points ykj are dense in S2(y) and y ∈U, we obtain a contradiction that yk ∈f(D). (2) If ˜x ∈X, then we can repeat (1) for D = Oδ1(˜x) in the case x ∈f −1(y), else if x ∈X \ f −1(y), we can repeat (1). □ Lemma 2. f is a closed function at every point of Y0 = Y \ S n Yn. Hence, f|X0 is a closed function onto Y0 = f(X0). Since the preimages of points are compact, the assertion of the lemma follows from the definition of the sets Xn in A. □ 4. Preservation of Borel class by clopen-LC functions in the Cantor set C Theorem 2. Let f : X →Y be a continuous, clopen-LC

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