Metrization in a Moore space

Abstract

The primary purpose of the study is to find new necessary and sufficient conditions for metrizations of a Moore space, as well as to give some new forms for the questions: Is a normal Moore space metrizable and is a pointwise paracompact normal Moore space metrizable? Also the study includes comparisons of a Moore space to some other topological spaces. The majority of the metrizability conditions come from the study of the other spaces and their relations to a Moore space instead of from direct research looking for metrizability conditions. The following conditions are found to be equivalent in a Moore space, X. 1. X is metrizable. 2. X is paracompact. 3. X is pseudo-metrizable. 4. X has a sigma-closure-preserving base. 5. X has a sigma-closure-preserving quasi-base. 6. X has a sigma-cushioned pair-base. 7. X is a Nagata Space. 8. There exists a sequence {G_i} from i=1 to infinity satisfying the definition of a Moore space such that G_(i+l)* <. G_i for each i, It is also shown that the following are sufficient conditions for metrizability in a Moore space. 1. X has the Lindelof property. 2. Every uncountable subset has a limit point. 3. X is normal, pointwise paracompact and separable. 4. For every open cover N of X, there exists an open cover M of X such that M < N where M is point finite and for each x, y in x, if x is in the intersection of {m|y in m in M} then y is in the intersection {m|x in m in M}. 5. There exists a perfect map, f, from X to a paracompact space Y. 6. {S(x, G_i)| x in X, i = 1, 2, . . .] is a uniformity, in the sense of Nagata, for X. Where G = {G_i} from i=1 to infinity is as in the definition of a Moore space. 7. G = {G_i} from i=1 to infinity is a uniformity, in the sense of Tukey, for X. Some equivalent conditions for countable pointwise paracompactness are proved. It is also found that in a Moore space normality implies countable paracompactness. Throughout the paper unanswered questions are proposed.