  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 $C_p(X)$.
