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 homeotopy group of $X$.

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

$MCG^{*}(X)=\pi _{0}(\operatorname {Homeo} (X),{\mbox{identity map}})=\operatorname {Homeo} (X)/({\mbox{Path component of identity in }}\operatorname {Homeo} (X))$