## 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.

## Recommended Citation

Pixley, Carl Preston, "Selection theory for infinite dimensional spaces and continuous single-valued approximations to upper semi-continuous multivalued mappings" (1972). *Graduate Dissertations and Theses*. 181.

https://orb.binghamton.edu/dissertation_and_theses/181