Document Type

Dissertation

Date of Award

1971

Keywords

Topology, Decomposition spaces, Separation properties

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematical Sciences

First Advisor

Louis F. McAuley

Second Advisor

Dick Wick Hall

Third Advisor

Hudson V. Kronk

Series

Science and Mathematics

Abstract

This thesis deals primarily with decomposition spaces and the question of inheritance by a decomposition space of certain topological properties. Some new topological concepts which are introduced are of independent interest but they are explored here principally for their implications in decomposition spaces.

In Chapter II we compare McAuley's definition of an upper semicontinuous decomposition with other separation properties of the decomposition space and relations of these properties to the projection map. In contrast to Whyburn's (originally, Moore's) definition of upper semicontinuity, which is equivalent to requiring the projection map to be closed, these are purely topological properties, but some nevertheless impose conditions on quotient maps onto spaces satisfying them. Also, they are investigated in conjunction with various basis restrictions on the decomposition space (such as first countability, etc.) or conditions on the nature of the individual elements of the decomposition.

Chapter III is more narrow in scope, dealing specifically with certain shrinkability theorems of McAuley, originally asserted for decompositions which are upper semicontinuous in the sense he defined. The observation that this definition of upper semicontinuity did not yield the desired properties as supposed led to the investigations of Chapter II. Proofs of the theorems with the amended hypotheses are supplied.

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