Connectedness im kleinen and local connectedness in 2^X and C(X)

Jack Tilden Goodykoontz, Jr.

Vol. 53 (1974), No. 2, 387–397

DOI: 10.2140/pjm.1974.53.387

Abstract

Let X be a compact connected metric space and 2^X(C(X)) denote the hyperspace of closed subsets (subcontinua) of X. In this paper the hyperspaces are investigated with respect to point-wise connectivity properties. Let M ∈ C(X). Then 2^X is locally connected (connected im kleinen) at M if and only if for each open set U containing M there is a connected open set V such that M ⊂ V ⊂ U (there is a component of U which contains M in its interior). This theorem is used to prove the following main result. Let A ∈ 2^X. Then 2^X is locally connected (connected im kleinen) at A if and only if 2^X is locally connected (connected im kleinen) at each component of A. Several related results about C(X) are also obtained.

Mathematical Subject Classification 2000

Primary: 54B20

Milestones

Received: 30 April 1973

Published: 1 August 1974