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Fasciculi Mathemat FASCICULIMATHEMATICI Nr 55 2015 DOI:10.1515/fascmath-2015-0024 Luong Quoc Tuyen MAPPING THEOREMS ON SPACES WITH sn-NETWORK g-FUNCTIONS Abstract. Let ∆ be the sets of all topological spaces satisfying the following conditions. (1) Each compact subset of X is metrizable; (2) There exists an sn-network g-function g on X such that if xn ! x and yn 2 g(n; xn) for all n 2 N, then x is a cluster point of fyng. In this paper, we prove that if X 2 ∆, then each sequentially- quotient boundary-compact map on X is pseudo-sequence-cove- ring; if X 2 ∆ and X has a point-countable sn-network, then each sequence-covering boundary-compact map on X is 1-sequence-covering. As the applications, we give that each sequentially-quotient boundary-compact map on g-metrizable spaces is pseudo-sequence-covering, and each sequence-covering boundary-compact on g-metrizable spaces is 1-sequence-covering. Key words: sn-networks, sn-network g-functions, g-metrizable spaces, boundary-compact maps, sequentially-quotient maps, pseudo-sequence-covering maps, sequence-covering maps, 1-se- quence-covering maps. AMS Mathematics Subject Classification: 54C10, 54E40, 54E99. 1. Introduction and preliminaries A study of images of topological spaces under certain sequence-covering maps is an important question in general topology. In 2001, S. Lin and P. Yan proved that each sequence-covering and compact map on metric spaces is 1-sequence-covering ([15]). Furthermore, S. Lin proved that each sequentially-quotient compact maps on metric spaces is pseudo-sequence- covering, and there exists a sequentially-quotient π-map on metric spaces is not pseudo-sequence-covering ([14]). In [1], T. V. An and L. Q. Tuyen proved that each sequence-covering π and s-map on metric spaces is 1-sequence-cove- ring. After that, F. C. Lin and S. Lin proved that each sequence-covering and 200 Luong Quoc Tuyen boundary-compact map on metric spaces is 1-sequence-covering ([10]). Re- cently, the authors proved that if X is an open image of metric spaces, then each sequentially-quotient boundary-compact map on X is pseudo-sequence- covering ([11]). Let ∆ be the sets of all topological spaces satisfying the following condi- tions. (1) Each compact subset of X is metrizable; (2) There exists an sn-network g-function g on X such that if xn ! x and yn 2 g(n; xn) for all n 2 N, then x is a cluster point of fyng. In this paper, we prove that if X 2 ∆, then each sequentially-quotient boundary-compact map on X is pseudo-sequence-covering; if X 2 ∆ and X has a point-countable sn-network, then each sequence-covering boundary- compact map on X is 1-sequence-covering. As the applications, we give that each sequentially-quotient boundary-compact map on g-metrizable spaces is pseudo-sequence-covering, and each sequence-covering boundary-compact on g-metrizable spaces is 1-sequence-covering. Throughout this paper, all spaces are assumed to be Hausdorff, all maps are continuous and onto, N denotes the set of all natural numbers. Let P be a collection of subsets of X, we denote S P = SfP : P 2 Pg. Definition 1. Let X be a space, fxng ⊂ X and P ⊂ X. (1) fxng is called eventually in P , if fxng converges to x, and there exists m 2 N such that fxg [ fxn : n ≥ mg ⊂ P . (2) fxng is called frequently in P , if some subsequence of fxng is even- tually in P . (3) P is called a sequential neighborhood of x in X [5], if whenever fxng is a sequence converging to x in X, then fxng is eventually in P . Definition 2. Let P be a collection of subsets of X. (1) P is point-countable, if each point x 2 X belongs to only countably many members of P. (2) P is locally finite, if for each x 2 X, there exists a neighborhood V of x such that V meets only finite many members of P. S (3) P is σ-locally finite, if P = fPn : n 2 Ng, where each Pn is locally finite. (4) P is a network at x in X, if x 2 P for every P 2 P, and whenever x 2 U with U open in X, then x 2 P ⊂ U for some P 2 P. (5) P is a cs-cover [19], if every convergent sequence is eventually in some P 2 P. Definition 3. Let fPn : n 2 Ng be a sequence of covers of a space X such that Pn+1 refines Pn for every n 2 N. Mapping theorems on spaces with . 201 S (1) fPn : n 2 Ng is a σ-strong network for X [8], if fst(x; Pn): n 2 Ng is a network at each point x 2 X. S (2) fPn : n 2 Ng is a σ-locally finite strong network consisting of cs-covers for X, if it is a σ-strong network and each Pn is a locally finite cs-cover. S Definition 4. Let P = fPx : x 2 Xg be a cover of a space X. Assume that P satisfies the following (a) and (b) for every x 2 X. (a) Px is a network at x. (b) If P1, P2 2 Px, then there exists P 2 Px such that P ⊂ P1 \ P2. (1) P is a weak base of X [2], if for G ⊂ X, G is open in X if and only if for every x 2 G, there exists P 2 Px such that P ⊂ G; Px is said to be a weak neighborhood base at x in X. (2) P is an sn-network for X [12], if each element of Px is a sequential neighborhood of x for all x 2 X; Px is said to be an sn-network at x in X. Definition 5. Let X be a space. Then, (1) X is gf-countable [2] (resp., snf-countable [7]), if X has a weak base S (resp., sn-network) P = fPx : x 2 Xg such that each Px is countable. (2) X is g-metrizable [17], if X is regular and has a σ-locally finite weak base. (3) X is sequential [5], if whenever A is a non closed subset of X, then there is a sequence in A converging to a point not in A. (4) X is strongly g-developable [18], if X is sequential has a σ-locally finite strong network consisting of cs-covers. Remark 1. (1) Each strongly g-developable space is g-metrizable. (2) A space X is gf-countable if and only if it is sequential and snf-coun- table. Definition 6. Let f : X −! Y be a map. (1) f is a compact map [4], if each f −1(y) is compact in X. (2) f is a boundary-compact map [4], if each @f −1(y) is compact in X. (3) f is a pseudo-sequence-covering map [8], if for each convergent se- quence L in Y , there is a compact subset K in X such that f(K) = cl(L). (4) f is a sequentially-quotient map [3], if whenever fyng is a convergent sequence in Y , there is a convergent sequence fxkg in X with each xk 2 −1 f (yn ). k S (5) f is a weak-open map [21], if there exists a weak base P = fPy : −1 y 2 Y g for Y , and for y 2 Y , there exists xy 2 f (y) such that for each open neighborhood U of xy, Py ⊂ f(U) for some Py 2 Py. (6) f is an 1-sequence-covering map [12], if for each y 2 Y , there is −1 xy 2 f (y) such that whenever fyng is a sequence converging to y in 202 Luong Quoc Tuyen −1 Y , there is a sequence fxng converging to xy in X with xn 2 f (yn) for every n 2 N. (7) f is a sequence-covering map [17], if every convergent sequence of Y is the image of some convergent sequence of X. Remark 2. (1) Each compact map is a compact-boundary map. (2) Each 1-sequence-covering map is a sequence-covering map. Definition 7 ([6]). A function g : N × X −! P(X) is called an weak base g-function on X, if it satisfies the following conditions. (1) x 2 g(n; x) for all x 2 X and n 2 N. (2) g(n + 1; x) ⊂ g(n; x) for all n 2 N. (3) fg(n; x): n 2 Ng is a weak neighborhood base at x for all x 2 X. Note that a weak base g-functions were called CWC-maps and CWBC-maps in [9] and [16], respectively. Definition 8. A function g : N × X −! P(X) is called an sn-network g-function on X, if it satisfies the following conditions. (1) x 2 g(n; x) for all x 2 X and n 2 N. (2) g(n + 1; x) ⊂ g(n; x) for all n 2 N. (3) fg(n; x): n 2 Ng is an sn-network at x for all x 2 X. 2. Main results Let ∆ be the sets of all topological spaces satisfying the following condi- tions. (1) Each compact subset of X is metrizable; (2) There exists an sn-network g-function g on X such that if xn ! x and yn 2 g(n; xn) for all n 2 N, then x is a cluster point of fyng.
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