For my topology class, I have to prove that if $X$ and $Y$ are compact Hausdorff and the graph of $f:X \to Y$ is closed, then $f$ is continuous. If $\{x_\lambda\}$is a net in $X$ that converges to $x$, I want to show that $\{f(x_\lambda)\}$ converges to $f(x)$. Since the graph of $f$ is a closed subset of a compact space, it is compact, and so the net $\{(x_\lambda,f(x_\lambda))\}$ has a convergent subnet in the graph. This subnet must converge to $(x,f(x))$, and so there is a subnet of $\{f(x_\lambda)\}$ converging to $f(x)$. I want to somehow show (possibly using the Hausdorff condition) that this implies that the whole net converges in $Y$, but I can't figure out how and I've been stuck for a while. Does anyone have any hints? I know there are other ways to prove it, but I want to prove it this way, especially since the theme of the homework is nets.
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1$\begingroup$ Try generalizing this lemma from sequences to nets. $\endgroup$ – Giuseppe Negro Oct 10 '14 at 10:08
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$\begingroup$ Thanks, that did it. That lemma seems useful, I'm surprised I've never heard of it before. $\endgroup$ – Liam Oct 10 '14 at 10:30
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You can use this lemma for the final part of the proof:
If $a$ is a cluster point of every subnet of a net $n$, then $n\to a$.
