Lemma 1

Let Z be a topological space, K a compact topological space and W an open subset of $Z\times K$. The set U of points $z\in Z$ such that $\{z\} \times K$ is contained in W is open in Z.

The first projection pr$_1: Z\times K\rightarrow Z$ is proper (TG, I, p. 77, cor. 5), hence closed. Consequently, $\complement U =$ pr$_1(\complement W)$ is closed in Z and U is open.