Vol. 53, No. 2, 1974

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

Jack Tilden Goodykoontz, Jr.

Vol. 53 (1974), No. 2, 387–397

Let X be a compact connected metric space and 2X(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 2X 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 2X. Then 2X is locally connected (connected im kleinen) at A if and only if 2X 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
Received: 30 April 1973
Published: 1 August 1974
Jack Tilden Goodykoontz, Jr.