Theorem 1

Let X, Y, Z be topological spaces, let $f: Z\rightarrow$ $X\times Y$ be an étale and separated mapping, and let $y_0$ be a point of Y. Let $s: X\times Y\rightarrow Z$ be a section of $f$. Suppose that the restriction of $s$ to $X\times \{y_0\}$ is continuous, as are analogously the restrictions of $s$ to $\{x\} \times Y$ for every $x\in X$. If the space Y is connected and locally connected (TG, I, p. 84, def. 4), the mapping $s$ is continuous.