Metrization of Symmetric Spaces and Regular Maps

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A symmetric d for a topological space R is said to be coherent if whenever $\{ x(n)\}$ and $\{ y(n)\}$ are sequences in R with $d(x(n),y(n)) \to 0$ and $d(x(n),x) \to 0$, then $d(y(n),x) \to 0$. V. Niemytzki and W. A. Wilson have essentially shown that a topological space R is metrizable if and only if R is symmetrizable via a coherent symmetric. Conditions on a symmetric d which are equivalent to d being coherent are established. As a consequence, a theorem of A. Arhangel’skiĭ may be refined by showing that if $f:R \to Y$ is a quotient map from a metrizable space R onto a ${T_0}$-space y, then Y is metrizable if and only if f is a regular map.