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 ) .