In this paper we show that two important generalized metric properties are generalizations of first countability. We give some conditions on these generalized metric properties which imply metrizability. We prove that, a space $X$ is metrizable if and only if $X$ is a strongly quasi-N-space, quasi$-gamma-$space; a quasi$-gamma$ space is metrizable if and only if it is a pseudo $wN-$ space or quasi$-$Nagata$-$space with quasi $G^*_gamma-$diagonal; a space $X$ is a metrizable space if and only if $X$ has a $CWBC-$map $g$ satisfying the following conditions: <p> 1. $g$ is a pseudo-strongly-quasi-N-map; <p> 2. for any $A subseteq X, overline{A} subseteq cup {g(n, x) : x in A}$. |
Keywords
Nagata space; $gamma-$ Space; metrizable; quasi$-G^*_gamma-$diagonal; first countable.
Math Review Classification
Primary AMS (1991) Subject Classification: 54E30, 54E35.
Last Updated
8 September 1999
Length
17 pages
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