## Tsukuba Journal of Mathematics

### 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.

#### Article information

Source
Tsukuba J. Math., Volume 28, Number 2 (2004), 455-471.

Dates
First available in Project Euclid: 30 May 2017

https://projecteuclid.org/euclid.tkbjm/1496164811

Digital Object Identifier
doi:10.21099/tkbjm/1496164811

Mathematical Reviews number (MathSciNet)
MR2105947

Zentralblatt MATH identifier
1131.57024

#### Citation

Banakh, T.; Cauty, R.; Trushchak, K.; Zdomsky, L. On universality of finite products of Polish spaces. Tsukuba J. Math. 28 (2004), no. 2, 455--471. doi:10.21099/tkbjm/1496164811. https://projecteuclid.org/euclid.tkbjm/1496164811