Self-homeomorphism group of locally connected locally compact Hausdorff space is a T0 topological group under the compact-open topology

Statement
Suppose $$X$$ is a locally connected locally compact Hausdorff space. Denote by $$\operatorname{Homeo}(X)$$ the self-homeomorphism group of $$X$$. Give $$\operatorname{Homeo}(X)$$ the compact-open topology (more precisely, the subspace topology arising from the compact-open topology on the space of all functions from $$X$$ to itself). Then, $$\operatorname{Homeo}(X)$$ becomes a topological group under this topology; in fact, it is a T0 topological group.