On $$\alpha $$ α -embedded subsets of products

Abstract

We prove that every continuous function $$f:E\rightarrow Y$$ 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 $$G_\delta $$ -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.