Homogeneous Subspaces of Products of Extremally Disconnected Spaces

TL;DR

This paper constructs specific homogeneous spaces with particular compactness properties and proves finiteness and metrizability of their compact subsets, advancing understanding of extremally disconnected spaces.

Contribution

It demonstrates that all compact subsets of homogeneous subspaces of finite powers of extremally disconnected spaces are finite, strengthening Frolík's theorem.

Findings

Constructed homogeneous countably compact spaces with non-pseudocompact products.

Proved all compact subsets of certain homogeneous subspaces are finite.

Under CH, all compact subsets of homogeneous subspaces are finite or metrizable.

Abstract

Homogeneous countably compact spaces $X$ and $Y$ whose product $X\times Y$ is not pseudocompact are constructed. It is proved that all compact subsets of homogeneous subspaces of the third power of an extremally disconnected space are finite. Moreover, under CH, all compact subsets of homogeneous subspaces of any finite power of an extremally disconnected space are finite and all compact subsets of homogeneous subspaces of the countable power of an extremally disconnected space are metrizable. It is also proved that all compact homogeneous subspaces of finite powers of an extremally disconnected space are finite, which strengthens Frol\'{\i}k's theorem.