On the t-equivalence relation

TL;DR

This paper investigates how certain topological and dimension properties of completely regular spaces are preserved under homeomorphisms and continuous open surjections of their associated function spaces $C_p(X)$, extending previous theorems.

Contribution

It strengthens a theorem of Okunev and derives new results on the preservation of dimension-type properties under mappings between function spaces.

Findings

Strengthened theorems on preservation of topological properties under homeomorphisms.

Proved new results on the preservation of dimension-type properties under continuous open surjections.

Extended previous work by Cauty and Marciszewski on function space mappings.

Abstract

For a completely regular space $X$, denote by $C_p(X)$ the space of continuous real-valued functions on $X$, with the pointwise convergence topology. In this article we strengthen a theorem of O. Okunev concerning preservation of some topological properties of $X$ under homeomorphisms of function spaces $C_p(X)$. From this result we conclude new theorems similar to results of R. Cauty and W. Marciszewski about preservation of certain dimension-type properties of spaces $X$ under continuous open surjections between function spaces $C_p(X)$.