On Homeomorphisms of Function Spaces Over Products with Compacta

TL;DR

This paper investigates the conditions under which homeomorphisms between certain function spaces imply similar homeomorphisms when extended to products with metric compacta, revealing structural invariances.

Contribution

It establishes that homeomorphisms between function spaces over products with compacta extend to products with some metric compactum, under countable netweight conditions.

Findings

Homeomorphism of $C_p(X\times Z)$ and $C_p(Y\times Z)$ implies similar for $C_p(X\times M)$ and $C_p(Y\times M)$

Existence of a metric compactum $M$ ensuring homeomorphism extension

Structural invariance of function spaces over product spaces with compacta

Abstract

We show that if $C_p(X\times Z)$ is homeomorphic to $C_p(Y\times Z)$, where $Z$ is compact, and $X$ and $Y$ are of countable netweight, then $C_p(X\times M)$ is homeomorphic to $C_p(Y\times M)$ for some metric compactum $M$.