| In this paper we investigate weak bases. We give a characterization of weakly developable spaces and metrization theorems. The metrization results are: a space $X$ is metrizable if and only if $X$ has a $CWBC$--map $g$ satisfying the following conditions: begin {enumerate} item $g$ is a pseudo--strongly--quasi--N--map; item for any $A subseteq X, overline {A} subseteq bigcup {g(n,x) : x in A }$; end {enumerate} a space $X$ is metrizable if and only if $X$ has a $CWBC$--map $g$ satisfying the following conditions: begin {enumerate} item if $x in g(n,y_n)$, $y_n in g(n,x_n)$, $x_n in g(n,y_n)$ and $y_n in g(n,x)$ for all $n in N$, then $x_n$ converges to $x$; item for any $A subseteq X, overline {A} subseteq bigcup {g(n,x) : x in A }$. end {enumerate} |
Keywords
weakly developable; metrizable; weakly first countable; quasi--$G^*_{delta}$-diagonal.
Math Review Classification
Primary 54E30, 54E35
Last Updated
Length
8 pages
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