proceedings of the american mathematical society Volume 72, Number 1, October 1978 STRATIFIABLE a-DISCRETE SPACES ARE M, GARY GRUENHAGE Abstract. It is shown that stratifiable o-discrete spaces are M¡. A corollary is that scattered stratifiable spaces are Mx. 1. Introduction. In 1961, J. Ceder [1] defined the Mrspaces, z = 1, 2, 3. Recently, H. Junnila [4] and the author [2] independently proved that the M3-spaces (i.e., stratifiable spaces) and the A/2-spaces are actually the same class of spaces. The main question, of course, is whether A/3-spaces are Mx, that is, whether every stratifiable space has a a-closure-preserving base. The only partial result in this direction is the author's proof in [2] that countable stratifiable spaces are A/,. Here we generalize this result to show that a-discrete stratifiable spaces are Mx. When combined with a result of P. Nyikos on scattered spaces, this proves also that scattered stratifiable spaces are A/,. But perhaps the main value of this result is the simplicity of the proof relative to the published proof for the countable case, which uses all the machinery of the author's proof that M3 -» M2. 2. Definitions and main results. A collection % of subsets of a_ space X is closure-preserving if whenever %' c%, then C1(IJ %') = U {ÎÎ\H E %'}. A space X is an A/, -space if it has a a-closure-preserving base of open sets. X is an M3-space (or stratifiable space) if, to each open U c X, one can assign_a sequence { U„}™=, of open subsets of X such that (a) U„ £ U, (h)u:.xun=u, (c) U„ c V„whenever U c V. A space X is monotonically normal [3] if for every pair of disjoint closed sets H and K, there exists an open set D(H,K) containing H such that D(H,K)r\K= 0, and if H' D H and K' c K, then D(H', K') D D(H,K). The only fact about stratifiable spaces used in the proof of our main result is that they are monotonically normal and hence collectionwise normal [3]. Theorem 1. A o-discrete stratifiable space is an Mx-space. Proof. Let X = U™=xFn, where X is stratifiable and each F„ is a closed Presented to the Society, January 6, 1978; received by the editors October 11, 1977 and, in revised form, November 8, 1977. AMS (MOS) subject classifications(1970). Primary 54E20. Key words and phrases. Stratifiable, Af, -space, monotonically normal. © American Mathematical Society 1978 189 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use