# A Note on Spaces with Locally Countable Weak-bases

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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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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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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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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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