I am self studying from the book of Rick Miranda "Algebraic Curves and Riemann Surfaces."

On page 41, the end of proof of Proposition 3.12 finishes with the statement:

Discrete subsets of compact spaces are finite

However, this is generally not true, as this shows. So is the proof wrong, or have I failed to take in account something, like for example that we are working with Riemann surfaces?

If the proof is wrong, is there any way to fix it?

  • $\begingroup$ It is indeed not true in general. It might be implied by the topology of the object you are working with (I am pretty sure this is true in the Zariski topology), or the author might mean closed subset. $\endgroup$ – Mees de Vries Oct 23 '16 at 17:50
  • 2
    $\begingroup$ Miranda means closed subset. $F^{-1}(y)$ is a closed subset, and then what you want holds. $\endgroup$ – hwong557 Oct 23 '16 at 17:52
  • $\begingroup$ @hwong557: So to be clear, you mean "discrete closed subsets of compact spaces are finite" is a true statement? $\endgroup$ – Will R May 13 '18 at 5:08
  • $\begingroup$ @WillR Yes, because you can get an open cover where each open contains a single point of the closed set, then get a finite subcover. $\endgroup$ – hwong557 May 14 '18 at 5:47
  • $\begingroup$ And of course, $F^{-1}(y)$ is closed, being the continuous pre-image of the singleton $\{y\},$ which is closed since the ambient space is Hausdorff. $\endgroup$ – Will R May 14 '18 at 7:36

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