A paracompact semi-metric space which is not an $M\sb{3}$-space

Abstract

E. A. Michael has shown that a regular Hausdorff space 5 is paracompact if and only if any of the following is true: (1) every open cover of 5 has a a-locally finite open refinement [8], (2) every open cover of 5 has an open ^-closure preserving refinement [9], or (3) every open cover of S has an open <r-cushioned refinement [lO]. The Nagata-Smirnov Theorem [12] or [14] (see also Bing's Theorem 3 in [l]) states that a P3-space with a <r-locally finite base is metrizable. In [3] Jack Ceder defines an Mi-space to be a P3-space with a cr-closure preserving base and an M3-space to be a Pi-space with a <r-cushioned pair base (see Definition 1 below). By Michael's theorems cited above both Miand Af3-spaces are paracompact, and in view of the Nagata-Smirnov Theorem, one might suspect that they would even be metrizable. In [3], however, Ceder shows that such is not the case; in fact Miand M3-spaces need not be first countable, and they need not be metrizable even if they are first countable. Ceder does show, though, that a first countable M3(and hence Mi-) space is a paracompact semi-metric space, and he raises the question: is this a characterization—i.e. is every paracompact semi-metric space an Af3-space (and perhaps even an Mi-space)? A negative answer is given to that question in this paper. Also it was called to my attention by Professor Michael that a question raised by C. Borges [2] is answered (in the negative) by the same example below, which is a cosmic space (the regular continuous image of a separable metric space [ll]) that is not an Af3-space. Definition 1. Let P be a collection of ordered pairs of subsets of the TV-space S such that, for each p = (pi, p2)EP, px is open and P1C.P2, and such that, for every xES and every neighborhood U of x, there is a pEP for which XEP1C.P2EU. Then P is called a pair base for 5. Moreover, P is called cushioned if, for every QQP,