# Infinite powers and Cohen reals

## TL;DR

The paper constructs a specific zero-dimensional separable metrizable space of Cohen reals demonstrating that all homeomorphisms of its countable product behave like coordinate permutations, providing insights into topological symmetry and open problems.

## Contribution

It presents a consistent example of a space of Cohen reals where all homeomorphisms of its countable product are essentially coordinate permutations, highlighting the sharpness of a known result.

## Key findings

- All homeomorphisms act like coordinate permutations almost everywhere.
- Permutation varies continuously across the space.
- Example clarifies an open problem in topology.

## Abstract

We give a consistent example of a zero-dimensional separable metrizable space $Z$ such that every homeomorphism of $Z^\omega$ acts like a permutation of the coordinates almost everywhere. Furthermore, this permutation varies continuously. This shows that a result of Dow and Pearl is sharp, and gives some insight into an open problem of Terada. Our example $Z$ is simply the set of $\omega_1$ Cohen reals, viewed as a subspace of $2^\omega$.