Another construction on a space is the group of all self-homeomorphisms , denoted If X is a locally compact, locally connected Hausdorff space then a fundamental result of R. Arens says that will in fact be a topological group under the compact-open topology.
Under the above assumptions, the homeotopy groups for are defined to be:
Thus is the extended mapping class group for In other words, the extended mapping class group is the set of connected components of as specified by the functor
- G.S. McCarty. Homeotopy groups. Trans. A.M.S. 106(1963)293-304.
- R. Arens, Topologies for homeomorphism groups, Amer. J. Math. 68 (1946), 593–610.