Compactness and connectedness in topological groups

Abstract

We study the dynamic interrelation between compactness and connectedness in topological groups by looking at the scale of various levels of connectedness through the looking glass of compactness and vice versa. More precisely, we are interested in measuring the gap between the connected component c( G) and the quasi-component q( G) of a compact-like group G. Neither local compactness nor countable compactness of G “can distinguish” between the properties: 1. (a) c( G) = 1, 2. (b) q( G) = l, 3. (c) G is zero-dimensional; in particular, always c( G) = q( G) for such a group G. Pseudocompactness together with minimality “cannot distinguish” between (b) and (c), but pseudocompactness together with total minimality “distinguishes” between (a) and (b). In the opposite direction, connectedness “cannot distinguish” between compactness and {countable compactness plus minimality} for Abelian groups of nonmeasurable size. We also discuss the role of connectedness for the question when topology or algebra alone determine the topological group structure of a compact-like group.