On universality of finite products of Polish spaces

Abstract

We introduce and study the $n$-Dimensional Perfect Homotopy Approximation Property (briefly $n$-PHAP) equivalent to the discrete $n$-cells property in the realm of $LC^{n}$-spaces. It is shown that the product $X\times Y$ of a space $X$ with $n$-PHAP and a space $Y$ with $m$-PHAP has $(n+m+1)$-PHAP. We derive from this that for a (nowhere locally compact) locally connected Polish space $X$ without free arcs and for each $n\geq 0$ the power $X^{n+1}$ contains a closed topological copy of each at most $n$-dimensional compact (resp. Polish) space.