In a previous paper [l] we gave necessary and sufficient conditions for a quotient space of a pseudo-metrizable space to be pseudometrizable. In this note we give a short proof of the corresponding theorem for preservation of complete regularity by quotient maps. The proof specializes in an obvious way to the pseudo-metric case and has the advantage that, unlike the proof in [l], it requires neither the use of uniformities nor the complicated construction of that paper. Moreover, we obtain an

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... teresting explicit definition of a pseudo-metric (or, in the complete regularity case, a defining family of pseudo-metrics) for the quotient space. For the most part the terminology here is standard. But we wish to make some things explicit. If p is a pseudo-metric for X, and if e>0, x£A, and A, BEX, then The topology on a space X defined by a family P of pseudo-metrics for X is the topology with {A^p.JxJl pEP, e>0, x£A} as subbase. (We do not require in the above definition that P separate points; so the topology generated by P need not be Hausdorff.) Recall that a topology on X is completely regular if and only if it can be defined by a family of pseudo-metrics. Theorem 1. Let / be a /unction /rom a completely regular space X onto a topological space Y, and suppose that Y has the quotient topology relative to/. Then the /ollowing assertions are equivalent: (1) Y is completely regular. (2) There exists a /amily P0 o/ pseudo-metrics defining the topology o/ X and a subbase S o/ the topology o/ Y such that /or each G£S there exists p£Po and a set {e(y, p) \ y EG} of positive real numbers satisfying (0 A7"«(,.P)[f-1[y]]C/-1[G],t/y£G, (ii) P(/-1b]-/-1[2]) = «(3', />)-«(*, P), *fy, 2GG. (3) There exists a family P0 of pseudo-metrics defining the topology of X such that the topology of Y is defined by the family Q= {qp \ p £P0}