Homeotopy group

Definition
Suppose $$X$$ is a locally connected locally compact Hausdorff space and $$k$$ is a positive integer. Denote by $$\operatorname{Homeo}(X)$$ the self-homeomorphism group of $$X$$, given the structure of a topological space via the compact-open topology. $$\operatorname{Homeo}(X)$$ becomes a T0 topological group under this topology (see here).

The $$k^{th}$$ homeotopy group of $$X$$, denoted $$HME_k(X)$$, is defined as the $$k^{th}$$ homotopy group of $$\operatorname{Homeo}(X)$$. Note that since $$\operatorname{Homeo}(X)$$ is a topological group, even the case $$k = 0$$ gives a group, and the case $$k \ge 1$$ gives an abelian group. Explicitly:

$$HME_k(X) = \pi_k(\operatorname{Homeo}(X);\mbox{identity map})$$

The special case $$k = 0$$ gives the group $$HME_0(X)$$. This group is also called the extended mapping class group of $$X$$ and is denoted $$MCG^*(X)$$.