Autohomeomorphism groups
Let \(H(X)\) denote the group of homeomorphisms of a topological space \(X\). The authors show that if \(M\) is a metric one-dimensional manifold with or without boundary, then the compact-open topology on \(H(M)\) is the unique minimum group topology (one contained in all \(T_1\) group topologies) on \(H(M)\). They note that they were informed by the referee that this result was also obtained, using different techniques, by \textit{D. M. Hayes} in her unpublished Ph.D. dissertation [Minimality of the special linear groups, University of North Texas, Ph.D. Dissertation (1997)]. A compact, connected, locally connected, metric space is referred to as a nice space. The authors show that for every compact, zero-dimensional, metric group \(G\), there exists a continuum of pairwise non-homeomorphic nice spaces whose homeomorphism groups, each with the compact-open topology, are all topologically isomorphic to \(G\). \textit{D. Evans} and \textit{P. R. Hewitt} in [Ann. Pure Appl. Logic 46, No. 2, 201-209 (1990; Zbl 0714.03030)] constructed two infinite, compact, zero-dimensional metric groups which are algebraically isomorphic but not isomorphic as topological groups. From this and the previous result, the authors deduce that there are nice spaces \(X\) and \(Y\) such that \(H(X)\) and \(H(Y)\) are infinite, algebraically isomorphic, and homeomorphic but not topologically isomorphic.
scientific article; zbMATH DE number 1917825
scientific article