On the t-equivalence relation

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 $C_p(X)$.

💡 Research Summary

The paper investigates the so‑called t‑equivalence relation between function spaces of the form (C_{p}(X)), where (X) is a completely regular space and (C_{p}(X)) denotes the set of all real‑valued continuous functions on (X) equipped with the topology of pointwise convergence. Two spaces (X) and (Y) are said to be t‑equivalent if their function spaces (C_{p}(X)) and (C_{p}(Y)) are homeomorphic. The main purpose of the work is twofold: first, to strengthen a theorem of O. Okunev concerning which topological properties of the underlying spaces are preserved under a homeomorphism of their function spaces; second, to extend dimension‑preservation results originally obtained by R. Cauty and W. Marciszewski from the setting of homeomorphisms to the more general situation of continuous open surjections between function spaces.

Strengthening Okunev’s theorem.
Okunev (1999) proved that if (C_{p}(X)\cong C_{p}(Y)) then (X) and (Y) share several basic properties such as being a (k)-space, σ‑compact, having countable tightness, or being Fréchet‑Urysohn. The present paper shows that the preservation list can be considerably enlarged. Using a detailed analysis of the standard basic open sets of (C_{p}(X)) – sets of the form \