Date of Award

1974

Degree Type

Report

Degree Name

Master of Science (MS)

Department

Mathematics and Statistics

First Advisor

L. D. Loveland

Abstract

Throughout the paper we consider the setting where f is a continuous function (a mapping) whose domain X and range Y are both Hausdorff spaces. Our object is to determine conditions on the map f which insure that when X has a certain topological property Q, then Y will also have property Q. For example, if X is metrizable, then it does not necessarily follow that Y is a metric space; but if f is a perfect map, then metrizability is preserved. Chapter III is devoted to the study of this metrizability problem. In particular, we present Frink's [ 2] characterization of metrizable spaces, and we use it to show that a closed map f preserves metrizability provided Y is either first countable or for each p∈Y, f-1(p)n has a compact frontier. This was apparently first observed by Stone [ 6].

From Stone's result and from the result that first countable is preserved by open mappings, it follows easily that metrizability is preserved when f is both open and closed. In this case we can even describe a familiar metric for Y; namely, if p, q∈Y then the metric σ for Y is given by σ(p, q) = d (f-1(p), f-1(q) ) where d denotes the Hausdorff distance. This result is due to Balanchandran [ 1]. The proof we present, however, differs from Balanchandran's since ours depends heavily on a previous theorem due to Wallace [ 7] where necessary and sufficient conditions are given for a decomposition G of a metric space into disjoint, nonempty closed sets to be continuous.

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