A Characterization of M1-Spaces

Abstract. In this paper, we prove the following theorem which gives a characterization of M1-spaces by g-function. Theorem. A space (X;) is an M1-space if and only if there exists a function g:!X! such that: 1: fxg = \ng(n; x). 2: y 2 g(n; x)) g(n; y) g(n; x). 3: There exists a collection U = [nUn of open sets in (X;) such that: (a). [fg(n; x) : x 2 Ung is regular open for each Un 2 U. (b). If H X is closed with y =2 H, then y =2 Cl([fg(n; x) : x 2 Ung) for some Un 2 Un with H Un