Discrete Ultrafilters and Homogeneity of Product Spaces

TL;DR

This paper introduces discrete ultrafilters and explores their properties, then examines the homogeneity of product spaces involving certain classes of spaces, showing that some products are not homogeneous under specific conditions.

Contribution

The paper defines discrete ultrafilters, introduces intermediate classes of spaces between $F$-spaces and $eta ext{ω}$-spaces, and proves non-homogeneity results for their products.

Findings

No product of infinite compact $ ext{R}_2$-spaces is homogeneous.

Under $rak d = rak c$, no product of $eta ext{ω}$-spaces is homogeneous.

Basic properties of discrete ultrafilters are established.

Abstract

An ultrafilter $p$ on $\omega$ is said to be discrete if, given any function $f\colon \omega \to X$ to any completely regular Hausdorff space, there is an $A \in p$ such that $f(A)$ is discrete. Basic properties of discrete ultrafilters are studied. Three intermediate classes of spaces $\mathscr R_1 \subset \mathscr R_2 \subset \mathscr R_3$ between the class of $F$-spaces and the class of van~Douwen's $\beta\omega$-spaces are introduced. It is proved that no product of infinite compact $\mathscr R_2$-spaces is homogeneous; moreover, under the assumption $\mathfrak d =\mathfrak c$, no product of $\beta\omega$-spaces is homogeneous.