Corollary 2

Let $X$ be locally compact, connected and locally connected, and suppose each point of $X$ has a neighbourhood which has a countable base. Let $Y$ be a Hausdorff space whose topology has a countable base, and let $p : X \to Y$ be a continuous mapping such that, for each $y \in Y$, $\overline{p}^{-1}(y)$ is a discrete subspace of $X$. Then the topology of $X$ has a countable base.

For each $x \in X$, let $V_x$ be a compact neighbourhood of $x$ in $X$ which has a countable base. It follows from $§ 9$, no. 4, Theorem 2, Corollary 2, that the set $\mathfrak{B}$ of the $V_x$ satisfies the conditions of Theorem 1, and we complete the proof as in Corollary 1.

Notice that, in this corollary, it can happen that the restriction of $p$ to an arbitrarily small neighbourhood $V$ of a point of $X$ is not a homeomorphism of $V$ onto $p(V)$.