Local connectivity in homeomorphism groups
George McCarty · Bulletin of the American Mathematical Society · 1961
Recently there has been increasing interest in the local connectivity of the group of all homeomorphisms of a manifold with boundary.Available tools of proof, however, seem to favor the case of a compact manifold [l ; 2] or else the use of the topology of uniform convergence of the group [3].The present note extends such results to the groups of homeomorphisms of certain noncompact manifolds, furnished with the compact-open topology (also see [4]).If X is a manifold with boundary, let G{X) be the group of homeomorphisms of X, with compact-open topology.G(X) is then a topological transformation group on X.Let the statement that a space X is (respectively) locally connected, locally contractible or locally w-connected be abbreviated by the phase U X is Ph", i=l, 2 or 3 respectively.