Document Type

Dissertation

Date of Award

1972

Keywords

Topological spaces, Topology, Selection theory, Infinite dimensional spaces, Continuous single-valued approximations, Semi-continuous multivalued mappings

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematical Sciences

First Advisor

Prabir Roy

Second Advisor

Louis F. McAuley

Third Advisor

Harry W. Berkowitz

Series

Science and Mathematics

Abstract

The extension problem is one of the fundamental problems of topology; if TX and Y are topological spaces and A is a closed subset of X and f:A —> Y is a continuous function, under what conditions is there »a continuous function fzx + Y such‘ that fIA = f ? Two key theorems of topology deal with this problem. They are the Tietze Extension A Theorem and the Brouwer No—Retraction Theorem. The first states that 1‘:-X is a normal Hausdorff space and Y is the real line then any continuous function f from a closed subset A of X into Y can be extended to a continuous function from X into Y . The second states that if X is the (n +1) -ball and Y and A are both the n-sphere bounding X in En+1 then the identity mapping from A to Y cannot be extended to a continuous function from X to Y.

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