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