On $\alpha$-embedded subsets of products

We prove that every continuous function $f:E\to Y$ depends on countably many coordinates, if $E$ is an $({\mathrm{\aleph }}_1,{\mathrm{\aleph }}_0)$-invariant pseudo-${\mathrm{\aleph }}_1$-compact subspace of a product of topological spaces and $Y$ is a space with a regular $G_{\delta }$-diagonal. Using this fact for any $\alpha <{\omega }_1$ we construct an $(\alpha +1)$-embedded subspace of a completely regular space which is not $\alpha$-embedded.