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I just started studying Topology this week so bear with me if the answer is trivial...

Given a topology $(X, \tau)$, where $X$ is an infinite set. Show that if all the infinite subsets of $X$ are closed, then $(X, \tau)$ is the discrete topology.

My approach is the following. Suppose that $S \subset X$. Then either $S$ is finite or infinite.

If $S$ is finite, then $X \setminus S$ is infinite (because $X$ infinite), and because $X \setminus S$ is closed, $S$ is open.

If $S$ is infinite, then I run into troubles. You see, take the natural numbers $X = \mathbb N$. Then $S = \mathbb N \setminus \{1 \}$ is infinite and its complement is finite. So there is no guarantee that $X \setminus S$ is infinite right? Do I miss something here?

  • Hint. A point is open (why?) – Hanul Jeon Jan 28 '18 at 10:45
  • @HanulJeon I have not encountered the definition of a point. It should be possible without I guess... –  Jan 28 '18 at 10:46
  • I meant a singleton. – Hanul Jeon Jan 28 '18 at 10:50
  • @HennoBrandsma: Is this really a duplicate of https://math.stackexchange.com/q/60269/42969? Here all infinite subsets are closed, and there all infinite subsets are open. – Martin R Jun 20 '21 at 11:06
  • @MartinR I said in my close vote that https://math.stackexchange.com/q/2624769 is the duplicate where the exact same question was asked. But that question was later marked duplicate with the different question (incorrectly). But transivity of duplicate wins here, I guess. – Henno Brandsma Jun 20 '21 at 11:08
  • @HennoBrandsma: What I mean is that you closed this question as a duplicate of https://math.stackexchange.com/q/60269/42969 (three years ago), and I wonder if that is correct. – Martin R Jun 20 '21 at 11:56
  • @MartinR that was not quite correct indeed. But I think there were more votes required at the time than just mine. So people must have seconded it. – Henno Brandsma Jun 20 '21 at 12:04

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You can see that if $x\in X$ is a point then $\{x\}$ is open, so every singleton is open and hence our space is discrete.

Hanul Jeon
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  • I'm sorry but I have to disagree about the marking of duplicate of this question...? How do you guys feel about that? –  Jan 28 '18 at 11:16
  • @Adrianos I agree with you, but reopening the question needs a vote. – Hanul Jeon Jan 28 '18 at 11:26