Problem with the definition of a discrete topology

Question

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?

@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. — , Sep 22 '17 at 22:15

score 12 · Accepted Answer · edited Sep 22 '17 at 17:34

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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.

edited Sep 22 '17 at 17:34

José Carlos Santos

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answered Sep 22 '17 at 17:32

Randall

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What in the world was wrong with writing $X-A$? – Randall Sep 22 '17 at 17:37
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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. – Thomas Andrews Sep 22 '17 at 17:39
@Randall Not only it is the standard notation as you can easily undo the edition. – José Carlos Santos Sep 22 '17 at 17:46
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I've taken to using \smallsetminus $X \smallsetminus A$. It makes both camps equally happy/unhappy. – Eric Towers Sep 22 '17 at 21:22

score 0 · Answer 2 · answered Sep 22 '17 at 17:43

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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.

answered Sep 22 '17 at 17:43

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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? – Paul Sinclair Sep 22 '17 at 19:17
@PaulSinclair Any idea concerning https://math.stackexchange.com/questions/3498846/whats-the-origin-of-the-term-chaotic-topology ? – Paul Frost Jan 08 '20 at 14:45
@PaulFrost - This (and now the other thread) are the only places where I have ever encountered this terminology, so I have no clue where it came from or how old it is. I find punctured dusk's speculation as to its origin to be weak, but do not fault them for posting it. – Paul Sinclair Jan 08 '20 at 14:51

Problem with the definition of a discrete topology

2 Answers2