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On homeomorphism groups and the compact open topology. (English) Zbl 1135.54023
If every point of a noncompact Hausdorff space $$X$$ has a neighbourhood which is a continuum then the compact-open topology on the space of homeomorphisms $$\mathcal H(X)$$ coincides with the topology inherited from the topological group of homeomorphisms of the one-point compactification of $$X$$. An example is given of a locally compact metric space $$X$$ for which the inverse operation on $$\mathcal H(X)$$ is not continuous.

##### MSC:
 54H11 Topological groups (topological aspects) 22A05 Structure of general topological groups 54C35 Function spaces in general topology 58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
##### Keywords:
space of homeomorphisms
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