A study of extremally disconnected topological spaces

Alexander Arhangel’Skii

doi:10.1007/s13373-011-0001-8

Abstract

We show that if an extremally disconnected space X has a homogeneous compactification, then X is finite. It follows that if a totally bounded topological group has a dense extremally disconnected subspace, then it is finite. The techniques developed in this article also imply that if the square of a topological group G has a dense extremally disconnected subspace, then G is discrete. See also Theorem 3.12. We also establish a sufficient condition for an extremally disconnected topological ring to be discrete (Theorem 3.9). A theorem on the structure of an arbitrary homeomorphism of an extremally disconnected topological group onto itself is proved (see Theorem 3.7 and Corollary 3.8).

Highlights

In this article, “a space” is “a Tychonoff topological space”
A space X is extremally disconnected if the closure of every open subset of X is open
We study extremally disconnected spaces from various points of view

Summary

Introduction

“a space” is “a Tychonoff topological space”. We consider various aspects of the theory of extremally disconnected spaces. Since every topological group is a homogeneous space, we immediately obtain from Theorem 2.6 the corollary: if a compact topological group G has a dense extremally disconnected subspace, G is finite. Theorem 2.9 Every totally bounded topological group G with a dense extremally disconnected subspace is discrete and is finite.

Results

Conclusion