Extended mapping class group

Definition

Suppose $X$ is a locally connected locally compact Hausdorff space. The extended mapping class group of $X$, denoted $MC{G}^{*}(X)$, is defined as the zeroth homeotopy group of $X$.

Explicitly, it is defined as follows: let $\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{o}(X)$ denote the self-homeomorphism group of $X$, viewed as a topological space with the compact-open topology. With this topology, it becomes a T0 topological group (see here). We define:

$MC{G}^{*}(X)={\pi }_0(\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{o}(X),\text{identity map})=\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{o}(X)/(\text{Path component of identity in }\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{o}(X))$