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Journal Article

A Note on Spaces with Locally Countable Weak-bases

TL;DR: In this paper, the authors established the relationships between spaces with a locally countable weak base and locally separable metric spaces by means of compact-covering maps, and showed that a regular space with a local weak base is g-metrizable.
Abstract: In this paper,firstly we establish the relationships between spaces with a locally countable weak-base and locally separable metric spaces by means of compact-covering maps,secondly we show that a regular space with a locally countable weak-base is g-metrizable.Thirdly,we show that 1-sequence-covering,quotient,ss-maps preserve spaces with a locally countable weak base.
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145 citations


"A Note on Spaces with Locally Count..." refers background in this paper

  • ...(3) f is a strong sequence-covering map ([11]) if each convergent sequence (including its limit point) of Y is the image of some convergent sequence(including its limit point) of X. (4) f is a sequence-covering map [ 5 ] if each convergent sequence(including its limit point) of Y is the image of some compact subset of X. (5) f is a …-map if (X,d) is a metric space and for each y 2 Y and its open neighborhood V in Y,d(fi1(y),X\fi1(V ......

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  • ...Remark 2.8. A compact-covering, quotient, compact image of a locally compact metric space 6) a space with a point-countable cs-network; see Example 9.8 in [ 5 ] or Example 2.9.27 in [10]....

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Journal ArticleDOI
01 Jan 1971
TL;DR: In this paper, it was shown that if X is an No-space and F is a paracompact N-space, then G(X, Y) with the compact-open topology is a ParacompACTN.
Abstract: A k-network (P for a space X is a family of subsets of X such that if CC U, with C compact and U open, then there is a finite union R of members of (P such that CCLRQ U. An Ha-space is a r3-space having a countable ¿-network and an it-space is a Tr space having a

104 citations


"A Note on Spaces with Locally Count..." refers background in this paper

  • ...A space X is an @-space if X has a ae-locally finite k-network ([ 21 ])....

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  • ...(1) P is a network X if for whenever x 2 V with V open in X, then x 2 P ‰ V for some P 2 P. (2) P is a k-network for X if for each compact subset K of X and its open neighborhood V , there exists a finite subfamily P0 of P such that K ‰ P0 ‰ V ([ 21 ])....

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Journal ArticleDOI
TL;DR: In this paper, the authors discuss the spaces containing a subspace having the Arens' space or sequential fan as its sequential coreflection, and some generalized metric spaces which contain no Arens's space and no sequential fan are studied.

66 citations


"A Note on Spaces with Locally Count..." refers background in this paper

  • ...is a sn-first countable space, and so Pfi is an fi4-space (see [ 13 ])....

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  • ...a sn-first countable space [ 13 ]) if X has a weak-base (resp....

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  • ...We have the following implications for a space X [23, 24, 13 , 3]....

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  • ...By Theorem 3.18 in [ 13 ], Pfi has a countable sn-network....

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  • ...For a family P of subsets of a space X and a map f : X ! Y , denote f(P) = {f(P) : P 2 P}. Readers can refer to [23, 13 ] for unstated definitions....

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Journal ArticleDOI
TL;DR: In this article, the authors investigated the structure of sequence-covering images of metric spaces, and showed that every sequence covering, quotient and s-image of a locally separable metric space is a local ℵ 0 -space.

43 citations


"A Note on Spaces with Locally Count..." refers background in this paper

  • ...By Corollary 4.7 in [ 16 ], there are a separable metric space Mfi and a compact-covering, quotient, compact map ffi from Mfi onto Xfi....

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