Vol. 56, No. 1, 1975

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ISSN: 0030-8730
Topologies compatible with homeomorphism groups

Peter Fletcher and Pei Liu

Vol. 56 (1975), No. 1, 77–86
Abstract

If 𝒜 is a class of open covers of a topological space (X,𝒯 ), then (X,𝒯 ) is said to be strongly 𝒜-stable provided that for each 𝒞∈𝒜 there is a homeomorphism h mapping X onto X, other than the identity homeomorphism, such that for each C ∈𝒞, h(C) = C. This paper studies strongly 𝒜-stable spaces. Although there are compact connected metric spaces that are not even strongly 𝒜-stable with respect to the class 𝒜 of all finite open covers, there is an extremely weak homogeneity condition that guarantees that a space (X,𝒯 ) is strongly 𝒜-stable with respect to the class .4 of all locally finite open covers. If H(X) is the full homeomorphism group of a space (X,𝒯 ) that is strongly 𝒜-stable with respect to the class 𝒜 of all finite open covers, then H(X) is nonabelian and there is a nondiscrete Hausdorff topology 𝒯 for H(X) such that (H(X),𝒯 ) is a topological group.

Mathematical Subject Classification 2000
Primary: 54H15
Milestones
Received: 16 October 1973
Revised: 4 April 1974
Published: 1 January 1975
Authors
Peter Fletcher
Pei Liu