Remark 1

Let B be a simply connected topological space and let $(E, p)$ be a covering of B. If the space E is connected and nonempty, the mapping $p$ is a homeomorphism. Let, for example, G be a connected topological group and H a discrete subgroup of G, so that the canonical mapping $p: G\rightarrow G/H$ makes G a covering of $G/H$ (I, p. 100, cor. 2 of th. 1). If the space $G/H$ is simply connected, the mapping $p$ is a homeomorphism and H is the one-element subgroup.