Functional equivalence of topological spaces

Let E be a non-trivial Banach space. The question when the spaces Cp(X,E) and Cp(Y,E) of all continuous mappings of X and Y into E in the topology of pointwise convergence are linearly homeomorphic is studied. These spaces are called lE-equivalent. A topological property or a cardinal function is called lE-invariant if it is preserved by the relation of the lE-equivalence. We prove that σ-discreteness, σ-scatteredness, the hereditary Lindelöf number, the hereditary density, the density, and the spread are lE-invariant properties. Moreover, we prove that in the class of μ-spaces of pointwise countable type the scatteredness, k-scatteredness, the extent, the paracompactness, and the p-paracompactness are lE-invariants. For that we introduce the notions of pm-equivalence, om-equivalence, pom-equivalence. We study some functional functors.