On $α$-embedded subsets of products

We prove that every continuous function $f:E\to Y$ depends on countably many coordinates, if $E$ is an $(\aleph_1,\aleph_0)$-invariant pseudo-$\aleph_1$-compact subspace of a product of topological spaces and $Y$ is a space with a regular $G_δ$-diagonal. Using this fact for any $α<ω_1$ we construct an $(α+1)$-embedded subspace of a completely regular space which is not $α$-embedded.

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