The closed image of a metrizable space is $M_{1}$

Abstract:

J. Ceder introduced the notions of ${M_1}$ space (a regular space with $\sigma$-closure preserving base) and stratifiable space as natural generalizations of Nagata and Smirnov’s conditions for the metrizability of a regular space. Even though a topological space Y which is the image of a metrizable space under a closed, continuous mapping need not be metrizable, we show as our main result that Y will have a $\sigma$-closure preserving base. It follows that one cannot obtain an example of a stratifiable space which is not ${M_1}$ by constructing a quotient space from an upper semicontinuous decomposition of a metric space. In the course of establishing our major result, we obtain conditions under which the image of certain collections of sets under a closed, continuous mapping will be closure preserving.