On Homeomorphism Groups and the Compact-Open Topology

If X is a topological space, then we let H(X) denote the group of autohomeomorphisms of X equipped with the compact-open topology. For subsets A and B of X we define [A, B] = {h ∈ H(X) : h(A) ⊂ B}, and we recall that the topology on H(X) is generated by the subbasis SX = {[K , O] : K compact, O open in X}. If X is a compact Hausdorff space, then H(X) is a topological group, that is, composition and taking inverses are continuous operations (see Arens [2]). It is well known that even for locally compact separable metric spaces the inverse operation on H(X) may not be continuous, thus H(X) is, in general, not a topological group (see the example to follow). However, it is a classic theorem of Arens [2] that if a Hausdorff space X is noncompact, locally compact, and locally connected, then H(X) is a topological group because the compact-open topology coincides with the topology that H(X) inherits from H(αX), where αX is the Alexandroff one-point compactification of X . We improve on this result as follows. (Recall that a continuum is a compact connected space. If A is a subset of X , then int A and ∂A denote the interior and boundary of A in X , respectively.)