Some Implications of Semi-1-connectedness1

A topological space X is said to be semi-re-connected at a point xGA" provided there exists an open set U containing x such that every Cech re-cycle on a compact subset of U bounds on X. (If X is regular, "a compact subset of" may be omitted.) If X is semi-re-connected at every point, then X is said to be semi-w-connected [l].2 As shown in [l], if a space has a finite 1-dimensional Betti number, it is semi-1connected, but the converse is not true. The purpose of this article is to prove certain "point set" properties implied by the hypothesis that the space is semi-1-connected. In what follows it is assumed that A" is a separable metric space. The coefficient group will be a field F.