A metrizable $X$ with $C_p(X)$ not homeomorphic to $C_p(X)\times C_p(X)$

TL;DR

This paper constructs an example of an infinite metrizable space where the space of continuous functions with the pointwise topology is not homeomorphic to its square, answering a longstanding open question in the theory of function spaces.

Contribution

It provides the first known example of such a space, demonstrating that $C_p(X)$ can differ topologically from its own square.

Findings

Constructed a zero-dimensional subspace of the real line as the example.

Showed $C_p(X)$ is not homeomorphic to $C_p(X) imes C_p(X)$ for this space.

Resolved a long-standing open problem in the theory of function spaces.

Abstract

We give an example of an infinite metrizable space $X$ such that the space $C_p(X)$, of continuous real-valued function on $X$ endowed with the pointwise topology, is not homeomorphic to its own square $C_p(X)\times C_p(X)$. The space $X$ is a zero-dimensional subspace of the real line. Our result answers a long-standing open question in the theory of function spaces posed by A.V. Arhangel'skii.