Properties and Examples Homeomorphism group

as other sets of maps between topological spaces, homeomorphism group can given topology, such compact-open topology. in case of regular, locally compact spaces group multiplication continuous.

if space compact , hausdorff, inversion continuous ,



h
o
m
e
o
(
x
)


{\displaystyle homeo(x)}

becomes topological group 1 can show. if



x


{\displaystyle x}

hausdorff, locally compact , locally connected holds well. there locally compact separable metric spaces inversion map not continuous ,



h
o
m
e
o
(
x
)


{\displaystyle homeo(x)}

therefore not topological group.

in category of topological spaces homeomorphisms, object groups homeomorphism groups.