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.

  • 1
    $\begingroup$ Try generalizing this lemma from sequences to nets. $\endgroup$ – Giuseppe Negro Oct 10 '14 at 10:08
  • $\begingroup$ 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

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 $n\to a$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.