A study of extremally disconnected topological spaces

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).