Theorem 1

Let $X$ be a topological space satisfying axiom $(O_{III})$ (but not necessarily Hausdorff), and suppose $X$ is connected and locally connected. Let $Y$ be a topological 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 $X$. Finally let $\mathfrak{B}$ be a set of subsets of $X$ whose interiors cover $X$ and such that: (i) The restriction of $p$ to each $V \in \mathfrak{B}$ is a closed mapping of $V$ into $Y$. (ii) Every $V \in \mathfrak{B}$ has a countable subset which is dense in $V$.

Then the space $X$ is the union of a countable family of open sets each of which is contained in a set of $\mathfrak{B}$.

Let $\mathfrak{B}$ be a countable base of the topology of $Y$. We shall say that a pair $(W, U)$ is distinguished if (i) $U \in \mathfrak{B}$ and (ii) $W$ is a component of $\overline{p}^{-1}(U)$ contained in a set of $\mathfrak{B}$.