Let $M$ be a locally compact (Hausdorff) space, and $g:M\to M$ an isomorphism (think of an action of a finite cyclic group).
By some generalities one can show that the "obvious" map $(M^g)^+\to(M^+)^g$ is continuous. Here $M^+$ is the one-point-compactification, and $M^g$ are the fixed points with the subspace topology. (One extends $g$ to a pointed map on $M^+$, that is $g(+)=+$).


Edit: if $M$ happens to be compact, $M^+=M\coprod +$ is the disjoint union with a point.


Is it true that this is an isomorphism? A (weak and/or equivariant) homotopy equivalence?
Is an appropriatly modified statement true in a "convenient" category of topological spaces?



Fixed points of one-point-compactification
$M^+$ is still Hausdorff, so also $(M^+)^g$ is. Now we observe that the "obvious" map is a continuous bijection from a compact to a Hausdorff space, thus by standard textbook contents, an isomorphism.


