Theorem 2

Let G be a discrete topological group. Let G operate on the right in $G^*$ by $h\cdot g=h\iota (g)$ and let B denote the quotient topological space $G^*/G$.

The space $G^*$ is a principal covering of B with group G; it is simply connected by paths.

The space B is a metrizable topological space, connected by paths, locally contractible, and its Poincaré group at every point is isomorphic to G.

Thus, the space B is a classifying space for the group G.

The group $G^*$ is contractible to its identity element (IV, p. 450, prop. 6). In particular it is connected by paths (III, p. 260) and simply connected by paths (IV, p. 340).

The group $\iota (G)$ is closed in $G^*$ (TG, III, p. 7, prop. 5), hence the space $G^*/G$ is metrisable (TG, III, p. 13, prop. 18 and TG, IX, p. 25, prop. 4). It is also connected by arcs (III, p. 258, prop. 3). Since the group $\iota (G)$ is discrete, it follows from corollary 2 of I, p. 100, that $G^*$ is a principal G-covering of B. The pointed covering $(G^*, e^*)$ is then a universal covering of the space B pointed at the image of $e^*$; the Poincaré group of B (at each of its points) is isomorphic to G.