On $\alpha$-embedded subsets of products

TL;DR

This paper investigates the properties of certain invariant subspaces in product spaces, proving a dependence on countably many coordinates for continuous functions and constructing examples of subspaces with specific embedding properties.

Contribution

It establishes conditions under which continuous functions depend on countably many coordinates and constructs examples of subspaces with distinct embedding levels.

Findings

Continuous functions depend on countably many coordinates in certain invariant subspaces.

Constructs of subspaces that are $(eta+1)$-embedded but not $eta$-embedded for $eta< ext{some ordinal}$.

Abstract

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 $G_\delta$-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.