The D-property of extremely normal spaces and the metrizability of certain compact spaces

Abstract

In this article, we show that every extremely normal space is a D-space. We introduce a notion called weakly extremely normal with respect to a weak base for a space X. We show that if a space X is weakly extremely normal with respect to a weak base for X, then X is a D-space. We finally show that a compact Hausdorff space X is metrizable if and only if X is sequential, every weak base for X is a k-network for X and X has property (σ-WA) with respect to a weak base for X. By this conclusion, we can get a known result that every compact Hausdorff space X with a point-countable weak base is metrizable.