Extending Continuous Functions
by
Phillip Zenor
Auburn University

Let C(X) denote the continuous real valued functions defined on X and let <C>(X) denote the set of all continuous functions such that the domain of f is a closed subset of X.

A function e:<C>(X)->C(X) is an extender if e(f) is an extension of f.

* An extender is monotone if whenever f and g are in <C>(X ) and have the same domain, then if f(x)>=g(x) for all x in their domain, them ef(x)>=eg(x) for all x in X.

* An extender is nested if whenever f and g are in <C>(X), the domain of f is a subset of the domain of g, and ef(x)>=g(x) for all x in the domain of g, then ef(x)>=eg(x) for all x in X.

We show that if X is monotonically normal and monotonically countably paracompact, then X admits a monotone extender. Locally compact LOTs and nestedly normal sigma-spaces admit nested extenders.

Date received: February 28, 2002