# 2017 seminar talk: On factorization properties of function spaces

Talk held by Witold Marciszewski (University of Warsaw, Poland) at the KGRC seminar on 2017-06-29.

### Abstract

For a Tychonoff space $X$, by $C_p(X)$ we denote the space of all continuous real-valued functions on $X$, equipped with the topology of pointwise convergence. One of the important questions (due to A.V. Arhangel'skii), stimulating the theory of $C_p$-spaces for almost 30 years and leading to interesting results in this theory, is the problem whether the space $C_p(X)$ is (linearly, uniformly) homeomorphic to its own square $C_p(X)\times C_p(X)$, provided $X$ is an infinite compact or metrizable space.

In my talk I will present some recent developments concerning these type of questions. In particular, I will show a metrizable counterexample to this problem for homeomorphisms. I will also show that, for every infinite zero-dimensional Polish space $X$, spaces $C_p(X)$ and $C_p(X)\times C_p(X)$ are uniformly homeomorphic.

This is a joint research with Rafal Gorak and Mikolaj Krupski.