$l_p(R)$-equivalence of topological spaces and topological modules

Let $R$ be a topological ring and $E$ be a unitary topological $R$-module. Denote by $C_p(X,E)$ the class of all continuous mappings of $X$ into $E$ in the topology of pointwise convergence. The spaces $X$ and $Y$ are called $l_p(E)$-equivalent if the topological $R$-modules $C_p(X,E)$ and $C_p(Y,E)$ are topological isomorphisms. Some conditions under which the topological property $\mathcal P$ is preserved by the $l_p(E)$-equivalence (Theorems 8–11) are given.