Local Finite Cohesion

Abstract

Local finite cohesion is a new condition which provides a general topological setting for some useful theorems. Moreover, many spaces, such as the product of any two nondegenerate generalized Peano continua, have the local finite cohesion property. If X is a locally finitely cohesive, locally compact metric space, then the complement in X of a totally disconnected set has connected quasicomponents; connectivity maps from X into a regular ${T_1}$ space are peripherally continuous; and each connectivity retract of X is locally connected. Local finite cohesion is weaker than finite coherence [4], although these conditions are equivalent among planar Peano continua. Local finite cohesion is also implied by local cohesiveness [l2] in locally compact ${T_2}$ spaces, and a converse holds if and only if the space is also rim connected. Our study answers a question of Whyburn about local cohesiveness.