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In wikipedia I've read that

the discrete topology on X is defined by letting every subset of X be open (and hence also closed), and X is a discrete topological space if it is equipped with its discrete topology

"hence also closed"? I couldn't get that part. If you let every subset of $X$ to be open, how come that makes them closed? I know that $\emptyset$ and $X$ are open (hence closed), called "clopen". But for a point $x \in X$ if I let it to be open in the topological space (I also didn't understand what "letting" procedure is, $x \in X$ is clearly closed, since it is a point), how come it becomes closed?

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    $\begingroup$ Points are neither open nor closed; only (sub)sets are. $\endgroup$ – jwodder Sep 22 '17 at 18:59
  • $\begingroup$ @jwodder: It is common to say that $x$ is a "closed point" when $\{ x \}$ is a closed set in the space. One does not often talk about "open points", but it should be clear that it means $\{ x \}$ is an open set. $\endgroup$ – Hurkyl Sep 22 '17 at 22:15
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$A \subseteq X$ is a closed because its complement $X\setminus A$ is still a subset of $X$, hence open.

This argument doesn't care how big $A$ is: it can be empty, a single point, the whole thing, or anything in between.

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    $\begingroup$ What in the world was wrong with writing $X-A$? $\endgroup$ – Randall Sep 22 '17 at 17:37
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    $\begingroup$ Randall, an edit isn't an insult. $X\setminus A$ is the standard notation. It is sometimes written $X-A$, but that can be ambiguous in certain contexts, so $\setminus$ is preferred. $\endgroup$ – Thomas Andrews Sep 22 '17 at 17:39
  • $\begingroup$ @Randall Not only it is the standard notation as you can easily undo the edition. $\endgroup$ – José Carlos Santos Sep 22 '17 at 17:46
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    $\begingroup$ I've taken to using \smallsetminus $X \smallsetminus A$. It makes both camps equally happy/unhappy. $\endgroup$ – Eric Towers Sep 22 '17 at 21:22
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The discrete topological spaces are exactly the ones where every point set $\{ x\}$ is open (take unions!)

And a nice fact: It's not always true that points are closed! For example, take the chaotic topology (only $\emptyset$ and the whole set) in any set with more than one point. We need a separation axiom ($T_1$) to say every point is closed.

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    $\begingroup$ Someone has decided that "indiscrete" and "trivial" were not sufficient names for this topology, so we should start calling it the "chaotic" topology as well? $\endgroup$ – Paul Sinclair Sep 22 '17 at 19:17

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