A non-CLP-compact product space whose finite subproducts are CLP-compact

TL;DR

The paper constructs a Hausdorff space with finite products being CLP-compact but the infinite product not, answering a question about the behavior of CLP-compactness in product spaces.

Contribution

It provides the first example of a Hausdorff space with finite powers CLP-compact but infinite powers non-CLP-compact, addressing an open question.

Findings

Finite products of the constructed spaces are CLP-compact.

Infinite product of the space is not CLP-compact.

Answers an open question by Steprns and
61ostak.

Abstract

We construct a family of Hausdorff spaces such that every finite product of spaces in the family (possibly with repetitions) is CLP-compact, while the product of all spaces in the family is non-CLP-compact. Our example will yield a single Hausdorff space $X$ such that every finite power of $X$ is CLP-compact, while no infinite power of $X$ is CLP-compact. This answers a question of Stepr\={a}ns and \v{S}ostak.