Local compactness and homeomorphisms of spaces of continuous functions

In this paper we prove that
1) the spaces $C_p(S)$ and $C_p(T)$ of all continuous functions in the topology of pointwise convergence are not linearly homeomorphic if $S,T$ are not locally compact metrizable while the derivation set $T^{(1)}$ is compact and the derivation set $S^{(1)}$ is not;
2) the spaces $C_K(X)$ and $C_K(Y)$ of all continuous functions in the compact-open topology are not homeomorphic if $X$ and $Y$ are completely regular spaces while $X$ is locally compact and $\sigma$-compact and there is a point $y_0\in Y$ of countable character such that every neighborhood of it is not a pseudocompact.