Everywhere dense subspaces of topological products and properties associated with final compactness

We prove the following. The $\sigma$–product of a family $\mathfrak{U}$ of topological spaces with countable base is a Lindelöf $\Sigma$-space if and only if $\mathfrak{U}$ has at most $2^{\aleph_0}$ non-homeomorphic elements. The $\sigma$-product of $\mathscr{K}$-analytical spaces is itself $\mathscr{K}$-analytical. Let $X$ be a $\sigma$-product of Lindelöf $\Sigma$-spaces and $C_p(X)$ the space of all continuous real-valued functions on $X$ in the topology of pointwise convergence. Then every bicompact $f\subset C_p(X)$ is a Frechet–Uryson space.