Document Type

Dissertation

Date of Award

1977

Keywords

Homeomorphisms

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematical Sciences

First Advisor

Louis F. McAuley

Second Advisor

Ross Geoghegan

Third Advisor

Patricia T. McAuley

Abstract

This dissertation consists of (A) a history of local homeomorphisms and (B) my research pertaining thereto.

In part (A), research concerning local homeomorphisms is traced from 1906—beginning with the work of Hadamard—up to the present. It is shown that a few major questions such as, “When is a local homeomorphism a homeomorphism?,” generated the major portion of the research activity.

In part (B), three major concepts are defined and/or considered, namely, H-connected spaces, induced decompositions, and locally separating sets.

We say that a T2 connected space X is H-connected iff any proper local homeomorphism of a connected space onto X is a homeomorphism. It is shown that H-connectedness is a legitimate generalization of simple connectedness in the context of locally path connected spaces, and properties of H-connected spaces are determined. For example, the union of two “nice” H-connected spaces is H-connected provided their intersection is connected.

It is proved that any proper local homeomorphism g of a 1° T2 space X onto a path connected space Y induces a decomposition X1,...,Xk of X into connected sets Xi such that gxi is a 1-1 map onto Y. Generalizations and refinements of this result are obtained.

We say that a closed subset M of a space X separates X locally iff there is an open set V ⊃ M such that V-M = A ∪ B where A and B are separated sets such that M ⊂ A ∩ B We prove that if X is any T2 space with a subset which separates X locally but not globally, then X has a connected k-fold covering space (X*,p) for each k which decomposes into k mutually disjoint connected sets Xi such that Pxi is a 1-1 map onto X; moreover, the sets Xi are mutually homeomorphic.

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