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.