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Graduate Seminar

Jing Zhang
Carnegie Mellon University
Title: What does it take to insert a continuous function in between?

Abstract: Let $X$ be any topological space and $f, g: X \to \mathbb{R}$ be lower semicontinuous and upper semicontinuous respectively with $f < g$. What topological axioms about $X$ are needed in order to insert (strictly) a continuous $h$ in between $f$ and $g$ (call this Insertion Property)? Try this pre-talk exercise: if $X$ is a metric space, this is true. We will discuss some finer analysis (due to C. H. Dowker) that this happens precisely when $X$ is normal and paracompact. A space that is normal but the Insertion Property does not hold is called Dowker space. We will collect some facts about Dowker spaces including their existence, which does not require axioms other than ZFC. However getting nicer ones remains mysterious so some interesting open questions will also be surveyed.

Date: Tuesday, March 7, 2017
Time: 5:30 pm
Location: Wean Hall 8220
Submitted by:  Yangxi Ou
Note: Video will not be recorded per the request of the speaker.