Date of Award
Master of Science (MS)
Mathematics and Statistics
L. D. Loveland
L. D. Loveland
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.
Hamm, Kay J., "Maintaining Topological Properties on the Brink of Destruction" (1974). All Graduate Plan B and other Reports. 1157.
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