Vol. 68, No. 2, 1977

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A characterization of solenoids

Charles Lemuel Hagopian

Vol. 68 (1977), No. 2, 425–435

Suppose M is a homogeneous continuum and every proper subcontinuum of M is an arc. Using a theorem of E. G. Effros involving topological transformation groups, we prove that M is circle-like. This answers in the affirmative a question raised by R. H. Bing. It follows from this result and a theorem of Bing that M is a solenoid. Hence a continuum is a solenoid if and only if it is homogeneous and all of its proper subcontinua are arcs. The group G of homeomorphisms of M onto M with the topology of uniform convergence has an unusual property. For each point w of M, let Gw be the isotropy subgroup of w in G. Although Gw is not a normal subgroup of G, it follows from Effros’ theorem and Theorem 2 of this paper that the coset space G∕Gw is a solenoid homeomorphic to M and, therefore, a topological group.

Mathematical Subject Classification
Primary: 54F20, 54F20
Received: 2 April 1976
Revised: 15 February 1977
Published: 1 February 1977
Charles Lemuel Hagopian