I was practicing for a qualifying exam and came across this bugger. Let me know your thoughts.
Question: Let $X, Y, Z$ be topological spaces, and let $f: X \times Y \to Z$ be a continuous function. If $U \subset Z$ is open, and $K \subset Y$ is compact, show that $\{ x \in X | f(x,y) \in U \mbox{ for all } y \in K \}$ is open in $X$.
Give an example to show that the condition that $K$ be compact cannot be removed.
My thoughts so far: If we restrict the domain of $f$, by saying $\tilde{f}:X \times K \to Z$ sends $(x,y) \to f(x,y)$, then this map remains continuous. Then we wish to look at the set of $x$ such that holding $x$ fixed yields a constant function in $y$, and show that this is open. Since $K$ is compact, any open covering has a finite subcovering... I'm not too sure where to go from here...
Any help would be much appreciated :)