Functional equivalence of topological spaces and topological modules

Let $R$ be a topological ring and $E$, $F$ be unitary topological $R$-modules. Denote by $C_pinsert ignore into journalissuearticles values(X,E);$ the class of all continuous mappings of $X$ into $E$ in the topology of pointwise convergence. The spaces $X$ and $Y$ are called $l_pinsert ignore into journalissuearticles values(E,F);$-equivalent if the topological $R$-modules $C_pinsert ignore into journalissuearticles values(X,E);$ and $C_pinsert ignore into journalissuearticles values(Y,F);$ are topological isomorphic. Some conditions under which the topological property $\mathcal{P}$ is preserved by the  $l_pinsert ignore into journalissuearticles values(E,F);$-equivalence insert ignore into journalissuearticles values(Theorems 6.3, 6.4, 7.3 and 8.1); are given.