On the t-equivalence relation

For a completely regular space X, denote by Cp(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 Cp(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 Cp(X).