Extended mapping class group

Definition
Suppose $$X$$ is a locally connected locally compact Hausdorff space. The extended mapping class group of $$X$$, denoted $$MCG^*(X)$$, is defined as the zeroth defining ingredient::homeotopy group of $$X$$.

Explicitly, it is defined as follows: let $$\operatorname{Homeo}(X)$$ denote the defining ingredient::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:

$$MCG^*(X) = \pi_0(\operatorname{Homeo}(X),\mbox{identity map}) = \operatorname{Homeo}(X)/(\mbox{Path component of identity in } \operatorname{Homeo}(X))$$