Submetrizable spaces

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1978

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Suppose X is a Moore space. It is known that if X is submetrizable, X has the j-link property for each positive integer j. If X admits a semimetric which is upper semi-continuous and continuous in one variable, then X has a v-normal development, a result due to H. Cook. We prove that if X is separable and X has a v-normal development, then X has the j-link property for each positive integer j. From this follow the corollaries: A Moore-closed space with a v-normal development is compact; and if X is a Moore space with a v-normal development then the closure of every conditionally compact subset of X is compact. We show that if j is an integer greater than 1, there is a Moore space which has the j-link property but not the j+l-link property. Alster and Przymusifiski have defined regular submetrizability and H. Cook has given conditions under which a regularly submetrizable Moore space admits a continuous semimetric. We introduce the stronger notion of normal submetrizability and show that a normally submetrizable space is completely regular. We also prove that if X[raised 2] is a Moore space, X is normally submetrizable, and the diagonal in X[raised 2] is closed in the metric topology, then X is continuously semimetrizable.

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