Weakly Chained Spaces

Abstract

We introduce "weakly chained spaces", which need not be locally connected or path connected, but for which one has a reasonable notion of generalized fundamental group and associated generalized universal cover. We show that in the compact metric case, weakly chained is equivalent to the concept of "pointed 1-movable" from classical shape theory. We use this fact and a theorem of Geoghegan-Swenson to give criteria on the metric spheres in a CAT(0) space that imply that the boundary is has semistable fundamental group at infinity.