Construction of functions with the given cluster sets

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1 arxiv: v1 [math.gn] 23 Feb 2016 Construction of functions with the given cluster sets Oleksandr V. Maslyuchenko Instytute of Mathematics Academia Pomeraniensis in Słupsk Słupsk, Poland Denys P. Onypa Department of Mathematical Analysis Chernivtsi National University Chernivtsi, Ukraine Abstract In this paper we continue our research of functions on the boundary of their domain and obtain some results on cluster sets of functions between topological spaces. In particular, we prove that for a metrizable topological space X, a dense subspace Y of a metrizable compact space Y, a closed nowhere dense subset L of X, an upper continuous compact-valued multifunction Φ : L Y and a set D X \L such that L D, there exists a function f : D Y such that the cluster set f(x) is equal to Φ(x) for any x L. 1 Introduction The notion of a cluster set was first formulated by Painleve [1] in the study of analytic functions of a complex variable. A fundamental research of cluster sets of analytic functions was carried out in [2]. Actually, the notion of cluster sets is a topological notion which characterizes a behavior of a function on the boundary of the domain of this function. The oscillation [3] is 2015 Mathematics Subject Classification: Primary 54C30; Secondary 54C10. Key words and phrases: cluster set, locally connected space. 1

2 2 O. Maslyuchenko and D. Onypa another limit characteristic of functions which is tightly connected with the notion of cluster sets. Namely, the oscillation is the diameter of the cluster set. The problem of the construction of functions with the given oscillation was investigated in [3, 4, 5, 6, 7, 8, 9, 10]. In [11] we researched the question on the limit oscillation of locally stable functions defined on open subsets of reals. In [12] we proved that for any upper semicontinuous function f : F [0;+ ], which is defined on the boundary F = G\G of some open subset G of a metrizable space X, there is a continuous function g : G R such that the oscillation ω g (x) is equal to f(x) for x F. In this paper we deal with the question about the construction of functions with the given cluster sets. Let F be a multifunction from X to Y. Recall, that the domain of F is domf = {x X : F(x) }. In the case when domf = D we write F : D Y. The graph grf of F is a subset of X Y which is defined by grf = {x} F(x). x X So, we have that domf is the projection of grf on X. Definition 1.1. LetX andy be topological spaces,d X andf : D Y. The cluster set of f at a point x X is called f(x) = U U x f(u D), where U x denotes the collection of all neighborhoods of x in X. The corresponding multifunction f we call the cluster multifunction of f. It is not hard to show that the graph grf of the cluster multifunction f is the closure of the graph grf of f. In general, D = domf domf D. Let Y be a compact space. By the Kuratowski theorem [13] the projection on X of any closed subset of X Y is closed in X. So, the domain of f is a closed subset of X as the projection of grf = grf. Then domf = D. It is easy to prove that f is continuous if and only if f(x) = {f(x)} for any x D. Moreover, if Y is a metric compact, then the oscillation ω f (x) of f at a point x D is equal to the diameter diam f(x) of the cluster set f(x). So, we deal with the following problem.

3 Construction of functions with given cluster sets 3 Problem 1.2. Let X be a topological space and Y be a dense subspace of a compact space Y, D X and L D \ D. Describe all multifunctions Φ : L Y such that there exists a function f : D Y with f(x) = Φ(x) for any x L. Proposition 3.2 and Theorem 4.3 give an answer on this problem in the case when X and Y are metrizable spaces and L is a closed nowhere dense subset of X. 2 σ-discrete sets Definition 2.1. A subset S of a space X is called discrete if for any x S there exists a neighborhood U of x in X such that U S = {x}; an ε-net if for any x X there exists y S such that d(x,y) < ǫ; ε-separated if s t X ε for any different points s,t S; separated if it is ε-separated for some ε > 0; σ-discrete if there exists a sequence of discrete sets S n such that S = S n. Proposition 2.2. Let X be a metrizable space and ε > 0. Then there exists an ε-separated ε-net S in X. Proof. Since the ε-separability is a property of finite character, using the Teichmüller-Tukey lemma [13, p.8] we build a maximal ε-separated set S. Then the maximality of S implies that S is an ε-net. Proposition 2.3. Let X be a metrizable space. Then there exist separated sets A n such that A = A n is a dense subset of X. In particular, the set A is σ-discrete and dense in X. Proof. Using Proposition 2.2 we pick an 1-separated 1-net A n n n for any n and put A = A n. Obviously, the set A is dense. Since every separated set is discrete, then A is σ-discrete as a union of separated sets.

4 4 O. Maslyuchenko and D. Onypa Proposition 2.4. Let X be a metrizable space and S X. Then the following statements are equivalent: (i) S is σ-discrete; (ii) there exists a sequence of separated sets S n such that S = (iii) for any infinitesimal sequence of positive numbers ε n there exists a sequence of sets S n such that S n is ε n -separated for any n and S = S n ; (iv) there exists a sequence of closed discrete sets S n in X such that S = S n. Proof. Prove that (i) implies (ii). Since S is σ-discrete, there is a sequence of discrete sets A n such that S = A n. Using Proposition 2.3 we pick separated setsa n,m such that m=1 S n ; A n,m is dense ina n. AsA n is a discrete set, so every dense set in it equals A n. Therefore, we have that A n = A n,m. m=1 Renumber N 2 = {(n k,m k ) : k N}. Put B k = A nk,m k and S k = B k \ B j j<k for any k N. The sequence (S k ) is desired. Prove that (ii) implies (iii). Let S = T n, where T n is δ n -separated. Fix an infinitesimal sequence of positive numbers ε n. Since ε n 0, there is n 1 such that ε n1 < δ 1. Analogically there is n 2 > n 1 such that ε n2 < δ 2. Continuing the process we have that for any k N there is n k > n k 1 such that ε nk < δ k. Thus, the set T k is ε nk -separated. Finally, put S n = T k if n = n k for some k N and S n = otherwise. Obviously, (iii) implies (iv), because every separated set is closed. The implication (iv) (i) is clear. Proposition 2.5. Let X and Y be metrizable spaces, Y be compact, M X Y be a σ-discrete set. Then a projection pr X M is σ-discrete. Proof. According to Proposition 2.4 we have that every σ-discrete set is a union of closed discrete sets. Thus, it is enough to show that projection of closed discrete set is closed and discrete. Let M be a closed discrete subset of X. Show that pr X M = S is closed and discrete. Let M be some subset of M. Since M is discrete and closed, the set M is closed in X Y. By the Kuratowski theorem [13, Theorem ] we conclude that S = pr X M is closed in X. Therefore, all subsets of S is closed in X. Thus, we have that S is closed and discrete.

5 Construction of functions with given cluster sets 5 3 Cluster sets of a function with a discrete domain For a metric space (X,d), a point a X, a nonempty set A X and ε > 0 we denote d(a, A) = inf d(a,x) and x A B(a,ε) = {x X : d(x,a) < ε}, B[a,ε] = {x X : d(x,a) ε}, B(A,ε) = {x X : d(x,a) < ε}, B[A,ε] = {x X : d(x,a) ε}. We also use more exact notations: B X (a,ε), B X (A,ε), B X [a,ε], B X [A,ε]. Proposition 3.1. Let X be a metrizable space, F be a nonempty closed subset of X, A be separable subset of X such that F A. Then there exist a sequence of points x n A such that F is equal to the set {x m : m n} of all limit points of (x n ). Proof. Without loss of generality we may assume that X = A. So, X is a separable metrizable space. Then by [13, Theorem 4.3.5] we can choose a totally bounded metric on X. Let S k be a finite 1 -net in X. Pick s S 2k k. Since A is dense in X, there is a s A such that d(x,a s ) < 1. Put 2k A k = {a s : s S k }. Then A k is a finite 1 -net in X. Consider finite sets k B k = A k B(F, 1 ). Let k B 1 = {x 1,...,x n1 },B 2 = {x n1 +1,...,x n2 },...,B k = {x nk 1 +1,...,x nk },... Show that the sequence (x n ) is desired. Since {x m : m > n k } = {x m : m > n k } B[F, 1 ]. So, k B j j>k B(F, 1 ), we have that k m=1 {x n : n m} = {x n : n n k } k=1 B[F, 1] = F. k Let x F. Then there exists y k A k such that d(x,y k ) < 1 k. Thus, y k A k B(F, 1 k ) = B k. Then there exists m k such that n k 1 < m k n k and y k = x mk. Therefore, (y k ) is a subsequence of (x n ) and y k x. So, x is a limit point of x n. Recall, that a multifunction Φ : X Y is called upper continuous if for any point x X and for any open set V with Φ(x) V there exists a neighborhood U of x in X with Φ(U) V. k=1

6 6 O. Maslyuchenko and D. Onypa Proposition 3.2. Let X be a topological space, Y be a compact, D X and f : D Y. Then f : D Y is an upper continuous compact-valued multifunction. Proof. For any x D we have that f(x) = f(u). So, f(x) is a closed U U x subset of a compact space Y. Therefore, f(x) is compact. Prove that f is upper continuous. Let x 0 D and let V be an open neighborhood of f(x 0 ). Put F = {f(u D) : U is a neighborhood of x 0 }. Then f(x 0 ) = F V. Since Y is compact, there exist F 1,F 2,...,F n F n such that F k V. Choose U k U x0 such that F k = f(u k ). Put U 0 = int n n k=1 k=1 k=1 U k. Then U 0 is a neighborhood of x 0 and f(u 0 D) n F k V. Therefore, for any x U 0 D we have that f(x) = f(u 0 ) V. So, f is upper continuous at x 0. k=1 f(u k ) = U U x f(u) Theorem 3.3. Let X be a metrizable space, Y be a dense subset of a metrizable compact Y, L be a closed nowhere dense subset of X, D = X \L and Φ : L Y be an upper continuous compact-valued multifunction. Then there is a set A such that A is discrete in D, A\A = L, and there exists a function f : A Y, such that f(x) = Φ(x) for any x L. Proof. Using Proposition 2.3 for a metrizable space grf we conclude that there exists a σ-discrete set M, which is dense in grf. Then M is σ-discrete in X Y too. By Proposition 2.5 we have that the projection S = pr X M is σ-discrete in X. Furthermore, L = pr X grφ pr X M pr X M = S. Then S is a dense subset of L. By Theorem 2.4 there exists a sequence of 3 -separated sets S n n such that S = S n. Set T n = S m for any n N. Now we construct families of points x s,k and y s,k where s S, k N, satisfying the following conditions: (1) x s,k D for any s S and k N; m<n (2) d(x s,k,s) < 1 n for any n N,s S n and k N; (3) x s,k s as k for any s S;

7 Construction of functions with given cluster sets 7 (4) x s,k x t,j if (s,k) (t,j); (5) y s,k Y for any s S and k N; (6) Φ(s) is the set of limit points of sequence (y s,k ) k=1 for any s S; (7) d(y s,k,φ(s)) < 1 n for any n N,s S n and k N. Fix s S 1 and put D s = B(s,1) D. Since D = X, s D s. Thus, s D s \D s. Therefore, there exists a sequence of different points x s,k D s, that tends to s. So, we have that conditions (1) (3) hold. Since a family of balls (B(s,1)) s S1 is discrete, then a family of sets (D s ) s S1 is discrete too. Thus, the condition (4) holds. Assume that points x s,k are constructed for some number n > 1 and for any s T n and k N, so that conditions (1) (4) hold. Fix s S n and put D s = B(s, 1 n ) D \{x t,j : t T n,j N}. Prove that s D s. Let A t = {x t,j : j N} for any t T n. Since x t,j t, A t = A t {t}. So s / A t for any t T n. For any m < n a family of balls (B(t, 1 )) m t S m is discrete. Besides, by the condition (2) we have that A t B(t, 1) if m < n and t S m m. Then a family (A t ) t Sm is discrete as m < n. Since T n = S m, we have that a family (A t ) t Tn is locally finite. m<n In this case, A t = A t s. t T n t T n So, a set U = X \ t T n A t = X \{x t,j : t T n,j N} is a neighborhood of s. Since D = X, s B(s, 1 n ) D. Thus, s B(s, 1 n ) D U = D s. Next, taking into account that s / D s, we have that s D s \ D s. Therefore, there exists a sequence of different points x s,k D s that tends to s. So, the conditions (1) (3) hold. Besides, x s,k x t,j as t T n, k,j N. Since a family of balls (B(s, 1 n )) s S n is discrete, we have that a family (D s ) s Sn is discrete too. Therefore, the condition (4) holds. Construct points y s,k. Fix n N and s S n. A set B Y (Φ(s), 1 n ) is an open neighborhood of a set Φ(s) in Y and Y is dense in Y. Then for a set Y s = B Y (Φ(s), 1 n ) we have that Y s = B Y (Φ(s), 1 n ) Y. Then Φ(s) Y s.

8 8 O. Maslyuchenko and D. Onypa Since every metrizable compact is a hereditarily separable, we have that the set Y s is separable. Using Proposition 3.1 with F = Φ(s) and A = Y s we conclude that there exists a sequence of points y s,k Y s such that Φ(s) is the set of limit points of (y s,k ) k=1. Therefore, the conditions (5) (7) hold. Put A = {x s,k : s S,k N} and check that A \ A = L. Firstly, by (3) we have that S A. Thus, L = S A. Secondly, by (1) we have that L A =. Thus, L A\A. Prove inverse inclusion. For any s S put A s = {x s,k : k N}. By (3) we have that A s = A s {s}. Since a family of balls (B(s, 1)) n s S n is discrete, a family (A s ) s Sn is discrete too. So, A s = A s = (A s {s}) = S n ( A s ) for every number n. s S n s S n s S n s S n Besides, by (2) we have that for any m n and s S m we have an inclusion A s B(s, 1) B[L, 1]. Hence, A m n s B[L, 1 ] for any number n. n m ns S m So, A = A s A s = A s A s m<n s S m m n s S m m<n s S m m n s S m (S m A s ) B[L, 1] B[L, 1] A. n n m<n s S m Therefore, A\A B[L, 1 ] for any n. Hence, A\A B[L, 1 ] = L. n n Define a function f : A Y by f(x s,k ) = y s,k for s S and k N and show that f is desired. Fix s 0 L and find out that f(s 0 ) = Φ(s 0 ). Prove that f(s 0 ) Φ(s 0 ). Pick ε > 0 and show that f(s 0 ) B(Φ(s 0 ),2ε). Since Φ is upper continuous, then there exists a neighborhood U 0 of s 0 X such that (8) Φ(U 0 L) B(Φ(s 0 ),ε). Choose δ > 0 such that B(s 0,2δ) U 0 and find a number n 0 such that 1 n 0 < δ and 1 n 0 < ε. Put U 1 = B(s 0,δ) and prove that (9) d(f(x),φ(s 0 )) < 2ε as x (A\T n0 ) U 1. Let x A\T n0 = S n such that x U 1. Then x = x s,k for some n n 0, n n 0 k N and s S n. Since x s,k U 1, then d(x s,k,s 0 ) < δ. Next, by (2) we have that d(x s,k,s) < 1 1 n n 0 < δ. Thus, d(s 0,s) < 2δ. So, s B(s 0,2δ) U 0. Therefore, s U 0 L. By (8) we have that Φ(s) B(Φ(s 0 ),ε). But f(x) = y s,k. By (7) we have that d(f(x),φ(s)) = d(y s,k,φ(s)) < 1 1 n n 0 < ε.

9 Construction of functions with given cluster sets 9 Thus, there is y Φ(s) such that d(f(x),y)) < ε. But d(y,φ(s 0 )) < ε, because y Φ(s) B(Φ(x 0 ),ε). Therefore, d(f(x),φ(s 0 )) < 2ε. So, (9) holds. As was remarked previously, (A s ) s Tn0 is a locally finite. Besides, by (3) we have that A s = A s {s}. Thus, by (4) we have that (A s ) s Tn0 is disjoint. So, we have that (A s ) s Tn0 is discrete. Let us show that there is a neighborhood U 2 U 1 of a point s 0 X such that (10) d(f(x),φ(s 0 )) < 2ε as x T n0 U 2. Firstly, consider the case when s 0 / T n0. Then for any t T n0 we have that s 0 / {t} A t = A t. Thus, s 0 / A t = A t. Then there is a t T n0 t T n0 neighborhood U 2 of s 0 such that U 2 A t = as t T n0. Therefore, the condition (10) holds. Now consider the case when s 0 T n0. Since s 0 A s0 {s 0 } = A s0 and a family (A s ) s Tn0 is discrete, we have that there is a neighborhood U 3 U 1 of a point s 0 such that U 3 A s = as s T n0 \{s 0 }. Next, by (6) we have that Φ(s 0 ) is a set of limit points of a sequence (y s0,k) k=1. Thus, there is k 0 N such that (11) d(y s0,k,φ(s 0 )) < 2ε for any k k 0. Put U 2 = U 3 \ {x s0,k : k < k 0 }. Then we have that U 2 T n0 {x s0,k : k k 0 }. Thus, (10) implies (11). Therefore, there is a neighborhood U 2 U 1 of s 0 such that a condition (10) hold. Thus, by (9) and (10) we have that d(f(x),φ(s 0 )) < 2ε as x A U 2. So f(a U 2 ) B(Φ(s 0 ),2ε). Hence, we have that f(s 0 ) = U U s0 f(a U) f(a U 2 ) B[Φ(s 0 ),2ε] for any ε > 0. Since Φ(s 0 ) is closed, we have that f(s 0 ) B[Φ(s 0 ),2ε] = Φ(s 0 ). ε>0 Let us find out that Φ(s 0 ) f(s 0 ). It is sufficient to show that Φ(s 0 ) f(u A) for any open neighborhood U of s 0. Fix some open neighborhood U 0 of s 0 and a point y 0 Φ(s 0 ). Show that y 0 f(u 0 A). Pick an open neighborhood V 0 of y 0 and show that V 0 f(u 0 A). Since (s 0,y 0 ) grφ = M, (U 0 V 0 ) M. Thus, there is a point

10 10 O. Maslyuchenko and D. Onypa (s,y) M (U 0 V 0 ). Then s pr X M = S, s U 0 and y Φ(s) V 0. Next, by (3) we have that there exists k 0 N such that x s,k U 0 as k k 0. By (6) we have that y is a limit point of a sequence (y s,k ) k=1. Thus, there exists k k 0 such that y s,k V 0. So, we have that x s,k U 0 A and f(x s,k ) = y s,k V 0 f(u 0 A). Hence, V 0 f(u A), and thus, y 0 f(u 0 A). 4 The main result Lemma 4.1. Let X be a metrizable topological space, Y be a dense subspace of a metrizable compact space Y, L be a closed nowhere dense subset of X, Φ : L Y be an upper continuous compact-valued multifunction and D X \L such that L D. Then there exists a function f : D Y such that f(x) Φ(x). Proof. Let x D and r(x) = d(x,l) = inf d(x,y) > 0. Then there exists y L h(x) L such that d(x,h(x)) < 2r(x). Since Y is dense in Y, there is a point f(x) B Y (Φ(h(x)),r(x)) Y. Let us show that f : D Y is desired. Let a L, ε > 0 and V = B Y (Φ(a), ε ). Since Φ is upper continuous, 2 then there is δ > 0 such that δ < ε and for a neighborhood U 0 = B X (a,δ) we have that Φ(U 0 L) V. Let U 1 = B X (a, δ). Pick x U 3 1 D. Then r(x) = d(x,l) d(x,a) < δ 2δ. Therefore, d(x,h(x)) < 2r(x) <. Thus, 3 3 d(a,h(x)) < d(a,x) + d(x,h(x)) < δ + 2δ = δ. Hence, h(x) U Besides, r(x) < δ < ε < ε. Thus, f(x) B Y (Φ(h(x)),r(x)) B Y (Φ(h(x)), ε). 2 Therefore, there exists y Φ(h(x)) such that d(f(x),y) < ε. Next, 2 y Φ(h(x)) Φ(U 0 D) V = B Y (Φ(a), ε ). Then there exists 2 z Φ(a) such that d(y,z) < ε. In this case, we have that d(f(x),z) < 2 d(f(x),y) + d(y,z) < ε. Thus, f(x) B Y (Φ(a),ε) for any x U 1 D. Hence, f(u 1 D) B Y (Φ(a),ε) B Y [Φ(a),ε]. Thus, f(a) = {f(u D) : U U a } f(u 1 D) B Y [Φ(a),ε] for any ε > 0. Therefore, f(a) B Y [Φ(a),ε] = Φ(a), because a set Φ(a) is closed. ε>0 Lemma 4.2. Let (M, ) be a directed set, (A m ) m M and (B m ) m M be decreasing sequences of sets. Then (A m B m ) ( A m ) ( B m ). Proof. Let x m M m M m M m M (A m B m ). Then for any m M we have that x A m B m. That is x A m or x B m. Let K = {k M : x A k } and L = M \K. Then for any l L we have that x B l.

11 Construction of functions with given cluster sets 11 Assume that K is cofinal. Then for any m M there is k K such that m k. If m M then m k for some k M and thus, x A k A m. Therefore, x A m for any m. Thus, x A m. Let K is not cofinal. Then m M there exists m 0 M such that for any k K we have that m 0 k. ( ) Check that a set L is cofinal. Pick m M. Since M is a directed set, there exists l M such that l m and l m 0. Then by ( ) we have that l / K. That is l M \K = L. So, L is cofinal. If m M then there exists l L such that m l and thus, x B l B m. Therefore, x B m. m M Theorem 4.3. Let X be a metrizable topological space, Y be a dense subspace of a metrizable compact space Y, L be a closed nowhere dense subset of X, Φ : L Y be an upper continuous compact-valued multifunction and D X \L such L D. Then there exists a function f : D Y such that f(x) = Φ(x) for any x L. Proof. By Theorem 3.3 we have that there exists a closed discrete set D 1 D and a functionf 1 : D 1 Y such thatd 1 \D 1 = L andf 1 (x) = Φ(x) as x L. Put D 2 = D \ D 1, L 2 = D 2 L and Φ 2 = Φ L2. Obviously, that L 2 is closed and nowhere dense and Φ 2 is an upper continuous compact-valued multifunction, moreover L 2 D 2. Thus, by Lemma 4.1 there exists a function f 2 : D 2 Y such that f 2 (x) Φ 2 (x) = Φ(x) as x L 2. Define a function f : D Y as follows: f(x) = { f1 (x), if x D 1, f 2 (x), if x D 2 ; for any x D = D 1 D 2. Check that f is desired. Fix x 0 D and show that f(x 0 ) = Φ(x 0 ). Firstly, let us find out that f(x 0 ) Φ(x 0 ). Indeed, taking into account that x 0 L D 1 and D 1 D we have that f(x 0 ) = {f(u D) : U U x0 } {f(u D 1 ) : U U x0 } = {f 1 (U D 1 ) : U U x0 } = f 1 (x 0 ) = Φ(x 0 ). Check that f(x 0 ) Φ(x 0 ). Consider the case when x 0 L\L 2. In this case we have that x 0 / D 2. Thus, there exists a neighborhood U 0 of x 0 such that U 0 D 2 =. Then for U U 0 we have that U D = U D 1. Therefore, f(x 0 ) = {f(u D) : U U x0 } = {f(u D) : U U 0,U U x0 }

12 12 O. Maslyuchenko and D. Onypa = {f(u D 1 ) : U U 0,U U x0 } = {f 1 (U D) : U U x0 } = f 1 (x 0 ) = Φ(x 0 ). Let x 0 L 2. Using Lemma 4.2 for a directed set (M, ) = (U x0, ), decreasing families A U = f(u D 1 ) and B U = f(u D 2 ), where U U x0, we have that f(x 0 ) = {f(u D) : U U x0 } = {f(u D 1 ) f(u D 2 ) : U U x0 } {f(u D 1 ) : U U x0 } {f(u D 2 ) : U U x0 } = {f 1 (U D 1 ) : U U x0 } {f 2 (U D 2 ) : U U x0 } = f 1 (x 0 ) f 2 (x 0 ) Φ(x 0 ). References [1] P. Painleve, Lecons sur la theorie analytique des equations differentielles professees a Stockholm 1895, Paris, [2] E. F. Collingwood and A. J. Lohwater, The theory of cluster sets, Cambridge Univ. Press, Cambridge, [3] P. Kostyrko Some properties of oscillation, Math. Slovaca 30 (1980), [4] J. Ewert, S. Ponomarev, On the existance of ω-primitives on arbitrary metric spaces, Math. Slovaca 53 (2003), [5] Z. Duszynski, Z. Grande and S. Ponomarev, On the ω primitives, Math. Slovaka 51 (2001), [6] S. Kowalczyk, The ω-problem, Diss. math. (Rozpr. mat.) 501 (2014), [7] Maslyuchenko O. V., Maslyuchenko V. K. The construction of a separately continuous function with given oscillation. Ukr. Math. J. 50 (1998), no. 7, (in Ukrainian) [8] Maslyuchenko O. V. The oscillation of quasi-continuous functions on pairwise attainable spaces // Houston Journal of Mathematics 35 (2009), no. 1,

13 Construction of functions with given cluster sets 13 [9] Maslyuchenko O. V. The construction of ω-primitives: strongly attainable spaces. Mat. Visn. Nauk. Tov. Im. Shevchenka 6 (2009), (in Ukrainian) [10] Maslyuchenko O. V. The construction of ω-primitives: the oscillation of sum of functions. Mat. Visn. Nauk. Tov. Im. Shevchenka 5 (2008), (in Ukrainian) [11] O. Maslyuchenko and D. Onypa, Limit oscillations of locally constant functions, Buk. Math. J. 1 (2013), no. 3-4, (in Ukrainian) [12] O. Maslyuchenko and D. Onypa, Limit oscillations of continuous functions, Carp. math. publ. (submitted for publication). [13] R. Engelking, General topology, Sigma Series in Pure Mathematics 6, Berlin, 1989.

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