Some generalizations of metric spaces

Abstract

In 1965, M. Wiscamb [30] studied two classes of generalized metric spaces called A1 and A2 spaces. She showed that Ai -spaces i = 1, 2 include the class of metrizable spaces and are contained in the class of collectionwise normal spaces. The purpose of this paper is to study these spaces, their properties, and introduce other intermediate spaces between A1 -spaces and the class of collectionwise normal spaces. Three classes of generalized metric spaces, A0 , 2-fold-fully normal, and almost-2-fold fully normal spaces are defined and their properties are studied. It is shown that an A0-space is completely normal and that the class of Ai-spaces possess all the properties of the class of-A0 -spaces. Some of the results are shown in the following implication diagram: " J. Ceder in [3] conjectured that a paracompact semi-metric space is an M3-space. However, this conjecture is false as R. W. Heath in [8] gave an example of a paracompact semi-metric space which is not an M3-space. It remains unknown, then, what condition on a semi-metric space gives an M3-space. Interesting enough, it is shown that every semi-metrizable Ai-space, i = 0, 1, 2 is an M3-space (Theorem 5.6). Finally, some properties of semi-metrizable Ai-spaces i = 0, 1, 2 are discussed and necessary and sufficient conditions for metrizability of Ai-spaces, i= 0, 1, 2 are given.