Corollary 3 (Poincaré-Volterra Theorem)

Let $Y$ be a locally compact, locally connected space whose topology has a countable base. Let $X$ be a connected Hausdorff space, and let $p : X \to Y$ be a continuous mapping which has the following property : for each $x \in X$ there is an open neighbourhood $U$ of $x$ in $X$ such that the restriction of $p$ to $U$ is a homeomorphism of $U$ onto an open subspace of $Y$. Then $X$ is locally compact and locally connected, and the topology of $X$ has a countable base.

Clearly $X$ is locally connected. Also each $x \in X$ has an open neighbourhood $U$ in $X$ such that the restriction of $p$ to $U$ maps $U$ homeomorphically onto an open subspace $p(U)$ of $Y$. Since $p(U)$ is a locally compact subspace of $Y$ ($§ 9$, no. 7, Proposition 13), there is a compact neighbourhood $W$ of $p(x)$ contained in $p(U)$, whence $U \cap \overline{p}^{-1}(W)$ is a compact neighbourhood of $x$ contained in $U$; thus $X$ is locally compact, since by hypothesis $X$ is Hausdorff. $U \cap \overline{p}^{-1}(W)$, being compact, is closed in $X$ ($§ 9$, no. 3, Proposition 4) and the conditions of Corollary 1 are therefore satisfied; hence the topology of $X$ has a countable base.