O. Karlova LIMITS OF SEQUENCES OF DARBOUX-LIKE MAPPINGS

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1 Математичнi Студiї. Т.40, 2 Matematychni Studii. V.40, No.2 УДК O. Karlova LIMITS OF SEQUENCES OF DARBOUX-LIKE MAPPINGS O. Karlova. Limits o sequences o Darboux-like mappings, Mat. Stud. 40 (2013), We call a mapping : X Y an l-darboux mapping i the image o any arcwise connected subset o X is connected. We prove that the class o l-darboux F σ -measurable mappings o a topological space to a metric space is closed with respect to uniorm limits. Е. Карлова. Пределы последовательностей отображений типа Дарбу // Мат. Студiї Т.40, 2. C Отображение : X Y мы называем l-дарбу отображением, если образ любого дугообразно связного подмножества пространства X связный. Доказано, что класс l-дарбу F σ -измеримых отображений между топологическим и метрическим пространствами замкнут относительно равномерных пределов. 1. Introduction. A mapping between topological spaces X and Y has the Darboux property i the image o any connected subset o X is connected. A mapping is a connectivity mapping i the graph o the restriction C is connected or every connected subset C o X. Notice that each connectivity mapping has the Darboux property. We say that is (weakly) Gibson ([4]) i (U) (U) or any open (connected) subset U o X. According to [3] each Darboux mapping is weakly Gibson. It is well known that the class o all Darboux Baire-one mappings : R R is closed with respect to uniorm limits ([1, Theorem 3.4]). It is naturally to ask whether the same is true or mappings deined on R n, n > 1. Observe that according to [2, Corollary 9.16] a uniorm limit o connectivity mappings m : R n R, n > 1, is a connectivity mapping. In the paper we consider l-darboux mappings or Darboux mappings in the sense o Pawlak (see deinitions in Section 3, [7]) and prove that the class o all l-darboux F σ -measurable mappings : X Y, where X is a topological space and Y is a metric space, is closed with respect to uniorm limits. To do this we irstly prove the auxiliary act that each weakly Gibson F σ -measurable mapping : X Y is a connectivity mapping i X is a connected subset o the real line and Y is an arbitrary space (see Section 2). It was proved in [6] that any unction : R R is a pointwise limit o Darboux mappings. In Section 4 we show that any mapping between an ω-resolvable space X and a separable space Y is a pointwise limit o Gibson mappings. 2. Connectivity unctions and G-closed sets. For a topological space X we denote by T (X) the topology o X, by C(X) we denote the collection o all connected subsets o X and by G(X) we denote the amily o all open connected subsets o X Mathematics Subject Classiication: 26A21, 26B05, 54C08. Keywords: uniorm limit; Darboux unction; F σ -measurable unction; weakly Gibson unction. c O. Karlova, 2013

2 LIMITS OF SEQUENCES OF DARBOUX-LIKE MAPPINGS 133 A mapping : X Y is a Baire-one mapping, B 1 (X, Y ), i there exists a sequence o continuous mappings n : X Y which is pointwise convergent to on X; F σ -measurable i 1 (V ) is an F σ -subset o X or any open set V Y ; (weakly) Gibson i or an arbitrary U T (X) (U G(X)) we have (U) (U); Darboux, D(X, Y ), i or any C C(X) the set (C) is connected; a connectivity mapping i the graph o C is connected or every connected subset C o X. A subset A o X is called ambiguous i it is both F σ and G δ in X. I : X Y is a mapping we deine γ : X X Y by γ (x) = (x, (x)) or all x X. I a space X is homeomorphic to a space Y we write X Y. Let A be a system o subsets o X. A set E X is called closed with respect to A or, briely, A-closed, i the inclusion A E implies A E or any A A. Here and throughout the paper we consider G-closed subsets o X, i.e. such sets which are closed with respect to the system o all open connected subsets o X. Let E X. Clearly, i E is closed or inte = then E is G-closed in X. Remark that the converse is not true. Indeed, let E = n=1 [ 1, 1 ]. Then E is a G-closed subset o R 2n 2n 1 with non-empty interior which is not closed. Theorem 1. Let X be a connected and locally connected hereditarily Baire space, X 1 be a G-closed ambiguous subset o X such that its complement X 2 is also G-closed. Then either X 1 = X or X 2 = X. Proo. Suppose on the contrary that X 1 X and X 2 X. Set F = X 1 X 2. Then F, because X is connected. We show that X 1 F is a dense set in F. To obtain a contradiction, assume that there exists x 0 F and an open connected neighborhood U o x 0 in X such that U F X 2. Then x 0 X 1 X 2, consequently, U X 1. Take an arbitrary a U X 1. It is easy to check that a X 2. Let G be a component o X \ X 2 such that a G. Then G is open in X, since X is locally connected. Notice that U G U \ G. According to [5, p. 136], U rg. Since G is closed in X \ X 2, rg X 2. Moreover, G X 1. Hence, rg F. Take any b U rg. Then b X 2. Since X 1 is G-closed, b G X 1, a contradiction. Thereore, the set X 1 F is dense in F. In the same manner we can see that the set X 2 F is also dense in F. Then X 1 F and X 2 F are disjoint dense G δ -subsets o F, which is impossible, because F is a Baire space. Hence, our assumption is alse. Thereore, X 1 = X or X 2 = X. Corollary 1. Let X be a connected subset o R and Y be a topological space. Then every weakly Gibson F σ -measurable mapping : X Y is a connectivity mapping. Proo. Remark that it is suicient to show that the graph o is connected. Assume it is not true. Then γ (X) = W 1 W 2, where W 1 W 2 = and W i is a non-empty clopen set in γ (X) or every i {1, 2}. Let X i = γ 1 (W i) or i {1, 2}. Then X = X 1 X 2 and X 1 X 2 =. We prove that the set X i is G-closed in X or i {1, 2}. Consider an open connected subset U o X such that U X 1. Then γ (U) γ (U). Indeed, let x 0 U and W be a neighborhood o γ (x 0 ) in X Y. Choose an open connected neighborhood U 0 o x 0 in

3 134 О. KARLOVA X and a neighborhood V o (x 0 ) in Y such that U 0 V W. Taking into account that U 0 U is an open connected subset o X, x 0 U 0 U and is weakly Gibson, we obtain that there exists x U 0 U such that (x ) V. Then γ (x ) U 0 V W. Hence, γ (U) γ (U) W 1 γ (X) = W 1. Thereor, U X 1. Analogously we can prove that X 2 is G-closed in X. Observe that the mapping γ is F σ -measurable. Indeed, let G be an open subset o X Y and {B k : k N} be a base o open sets in X. Put V k = {V : V is open in Y and B k V G}. Then W = k=1 B k V k. Since γ 1 (G) = k=1 (B k 1 (V k )), the set γ 1 (G) is F σ in X. Since W i is a clopen set, X i is an ambiguous subset o X or i {1, 2}. By Theorem 1 we have that either X 1 = X or X 2 = X. Then either W 1 = or W 2 =, which implies a contradiction. 3. Uniorm limit o l-darboux unctions. We call a mapping : X Y an l-darboux mapping i (C) is connected or any arcwise connected set C X. The collection o all l-darboux mappings between X and Y we denote by D l (X, Y ). Evidently, each Darboux mapping is l-darboux. It is not hard to veriy that D(R, Y ) = D l (R, Y ). But i the domain space is more complicated the inclusion D l (X, Y ) D(X, Y ) might to be strict (see Example 1). Proposition 1. Let X and Y be topological spaces. Then D l (X, Y ) i and only i or any set K X with K [0, 1] the set (K) is connected. Proo. Clearly, we only need to prove the suiciency. Let A be an arcwise connected subset o X. Fix an arbitrary a A. For all x A we consider a homeomorphic embedding ϕ x : [0, 1] A such that ϕ x (0) = a and ϕ x (1) = x. Write K x = ϕ x ([0, 1]). Then (A) = x A (K x). Since K x [0, 1], the set (K x ) is connected. Thereore, (A) is connected, provided (a) x A (K x). Example 1. For all (x, y) R 2 deine (x, y) = { sin 1 x, x > 0, 1, x 0. Then the mapping γ is l-darboux and is not Darboux. Proo. Let K R 2 be such a set that K [0, 1]. It is easy to check that D(R 2, R). Then g = K D(K, R). By Corollary 1, g is a connectivity unction. In particular, g has a connected graph. Since γ (K) is equal to the graph o g, γ (K) is connected. Thereore, γ is l-darboux by Proposition 1. Consider the set C = {(x, y) R 2 : y = sin 1, x > 0} {0, 0}. It is not hard to veriy that x the set γ (C) is disconnected. Hence, γ does not satisy the Darboux property. Let (X, d) be a metric space, x 0 X and ε > 0. By B(x 0, ε) we denote the set {x X : d(x, x 0 ) < ε}. Theorem 2. Let X be a topological space, (Y, d) be a metric space, ( n ) n=1 be a uniormly convergent sequence o (weakly) Gibson mappings n : X Y and let = lim n. Then is (weakly) Gibson.

4 LIMITS OF SEQUENCES OF DARBOUX-LIKE MAPPINGS 135 Proo. Take an arbitrary open (connected) set U X, a point x 0 U and ε > 0. Choose a number N so that d( N (x), (x)) < ε/2 or all x X. In particular, N (x 0 ) B((x 0 ), ε/2). Since N is (weakly) Gibson, there exists x 1 U such that Then N (x 1 ) B((x 0 ), ε/2). d((x 1 ), (x 0 )) d((x 1 ), N (x 1 )) + d( N (x 1 ), (x 0 )) < ε/2 + ε/2 = ε. Hence, (x 1 ) B((x 0 ), ε). Thereore, (x 0 ) (U). Theorem 3. Let X be a topological space, (Y, d) be a metric space and ( n ) n=1 be a uniormly convergent sequence o F σ -measurable mappings n D l (X, Y ). Then = lim n is an F σ -measurable l-darboux mapping. Proo. Assume that D l (X, Y ) and choose a subset K [0, 1] o X so that (K) = W 1 W 2, where W 1 and W 2 are disjoint non-empty clopen subsets o Z = (K). Let K i = 1 (W i ) or i {1, 2}. Set g = K and g n = n K or all n N. Show that g n : K Z is weakly Gibson or every n. Let U K be an open connected set, x 0 U and let V be an open neighborhood o (x 0 ) in Y. Suppose that (U) V =. Denote C = U {x 0 }. Then (C) = (U) {(x 0 )} (Y \ V ) {(x 0 )}, which contradicts the connectedness o C. According to Theorem 2 the mapping g : K Z is weakly Gibson. Then K 1 and K 2 are G-closed subsets o K. Moreover, g is F σ -measurable as a uniorm limit o F σ -measurable mappings ([5, p. 395]). Thereore, K 1 and K 2 are ambiguous subsets o K. Hence, K 1 = K or K 2 = K by Theorem 1. Consequently, W 1 = or W 2 =, a contradiction. 4. Pointwise limit o Gibson unctions. Recall that a topological space X is ω-resolvable i there exists a sequence (X n ) n=1 o dense subspaces o X such that X n X m = or n m and X = n=1 X n. Theorem 4. Let X be an ω-resolvable space and Y be a separable space. Then any mapping : X Y is a pointwise limit o sequence o Gibson mappings. Proo. Let (X n ) n=1 be a sequence o dense subspaces o X such that X n X m = or n m and X = n=1 X n, and let D = {y n : n N} be a dense subspace o Y. For all n N and x X deine { (x), i x X 1 X n, n (x) = y k, i x X n+k or some k. Fix n N and show that n : X Y is Gibson. Indeed, take an open set U X, a point x 0 U and a neighborhood V o n (x 0 ). Choose a number N so that y N V. Since X N+n is dense in X, there is a point x U X N+n. Then y N = n (x). Hence, n (U) n (U). It can be easily seen that lim n (x) = (x) or each x X. Remark 1. Let us consider the unction : R {0, 1}, = χ {0}. According to the previous theorem is a pointwise limit o a sequence o Gibson unctions. But is not a pointwise limit o Darboux unctions, because every Darboux unction h: R {0, 1} is constant.

5 136 О. KARLOVA REFERENCES 1. A. Bruckner, Dierentiation o real unctions, 2nd ed., Providence, RI: American Mathematical Society, 1994, 195 p. 2. R. Gibson, T. Natkaniec, Darboux-like unctions, Real Anal. Exchange, 22 ( ), 2, O. Karlova, V. Mykhaylyuk, On Gibson unctions with connected graphs, Math. Slovaca, 63 (2013), K. Kellum, Functions that separate X R, Houston J. Math., 36 (2010), K. Kuratowski, Topology, V.1, M.: Mir, (in Russian) 6. A. Lindenbaum, Sur quelques propriétiés des onctions de variable réelle, Ann. Soc. Math. Polon., 6 (1927), H. Pawlak, R.J. Pawlak, On weakly Darboux unctions and some problem connected with the Morrey monotonicity, Journal o Applied Analysis, 1 (1995), 2, Yuriy Fedkovych Chernivtsi National University Maslenizza.ua@gmail.com Received Revised