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I was doing problems from Topology Without Tears and found this question.

I have a topological space $(X,\tau)$ such that all its subsets are closed. I need to show that $(X,\tau)$ is a discrete space.

I know that all the subsets of $X$ are clopen and subset $\phi$ is also clopen. Can this fact be used to prove the same?

Henno Brandsma
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The Doctor
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  • If all subsets are closed, then their complements, i.e., again all subsets, are open. – Hagen von Eitzen May 26 '17 at 06:30
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    So, that implies that all the subsets of $X$ belong to $\tau$ and so we are done? Am I right? – The Doctor May 26 '17 at 06:32
  • Doesn't it say every infinite subset is closed? – Henno Brandsma May 26 '17 at 06:35
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    How you prove it depends a lot on how discrete topology is defined. One possible definition would be "all subsets are closed". I guess that's not the one you're using. – bof May 26 '17 at 06:43
  • @HagenvonEitzen The OP says he already knows that "all the subsets of $X$ are clopen". So, unless he has a very unusual definition of "clopen", it seems that he already knows that all subsets are open. – bof May 26 '17 at 06:46
  • You know that all the subsets of $X$ are clopen, and you know that the subset $\phi$ is also clopen. But what about the subset $X$ itself? Do you know that $X$ is clopen? – bof May 26 '17 at 06:48

3 Answers3

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Let $U \subseteq X$. We know that $X - U$ is closed by assumption, so $U$, being the complement of a closed set, is open. Since $U \subseteq X$ was arbitrary, then every subset $U$ of $X$ is open, and therefore $X$ is discrete.

C Squared
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You "know that all the subsets of $X$ are clopen". In particular they are all open, which is the definition of "$(X,\tau)$ is a discrete space".

Anne Bauval
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Let { Uα : αϵI } be collection of all subsets of set X. since all subsets of X are closed ⇒ compliments of all subsets are open ⇒ compliments of all subsets are in τ i.e U'α ϵ τ for each α ⇒ all subsets of x are in τ i.e Uα ϵ τ , so (X,τ ) is discrete topological space

user448047
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