I am working on this problem today:
Let $G=\{(x,f(x))|x\in X\}$ be the graph of the function $f:X\to Y$, where $X$ and $Y$ are topological spaces and $Y$ is compact. If $G$ is closed in $X\times Y$, show that $f$ is continuous.
The hint of the problem is to use the Tube lemma somewhere. If I let $U$ be an open set in $Y$, $x\in f^{-1}(U)$, and $V$ be an open subset in $X\times Y$ that contains $\{x\}\times Y$, then I cannot guarantee that the neighborhood $V_x$ such that $V_x\times Y\subset W$ is entirely inside $f^{-1}(U)$. Can someone tell me how I could use the tube lemma?
I feel like the solution in another post to the same problem does not really use the tube lemma.
