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I have come across the statement:

"A set is closed if and only if it contains all of its limit points".

on MSE. The limit point (in my understanding) of a set $P\subset X$ is a point $p\in X$ whose every neighborhood contains a point in $P$ other than $p$ itself. With this definition the singleton, $\{p\}$, has no limit points. Singletons are closed (in the usual topology in $\mathbb{R}$. Why is this not a contradiction?

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    $\begingroup$ What is your definition of a "limit point"? Isn't $x$ a limit point of $\{x\}$? $\endgroup$
    – Xander Henderson
    Feb 4 at 3:45
  • $\begingroup$ It depends. The definition in Baby Rudin says it isn't. $\endgroup$
    – MJD
    Feb 4 at 3:48
  • $\begingroup$ @MJD Which is why I asked what the definition being used here is... $\endgroup$
    – Xander Henderson
    Feb 4 at 3:49
  • $\begingroup$ While we're at it, what is your definition of "closed"? $\endgroup$
    – Xander Henderson
    Feb 4 at 3:52
  • $\begingroup$ @curiosity This question has already been asked and answered before on the site. See, for example, math.stackexchange.com/q/3828923 $\endgroup$
    – Xander Henderson
    Feb 4 at 4:00

1 Answer 1

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If a set has no limit points, then as your title suggests, it is vacuously satisfying the definition of a closed set. So there is no contradiction in this case.

The important point here, which many mathematicians take for granted but which we sometimes forget can be a little bit hard for others to swallow, is that in math, “vacuous” truth is not considered any less true than any other kind of truth.

When we say $A$ is defined to be closed if it “contains all its limit points” that logically translates into “For all limit points $x$ of $A$, $x\in A$.”

Crucially, in math, every “for all” statement is treated as only an assertion that there are no counterexamples, so the latter statement translates to “There are no limit points of $A$ which fail to lie in $A$.” In the case that $A$ is a singleton (in $\mathbb R$, or more generally, in a metric space, or even more generally, a so-called $T_1$ space), we can see that there are no limit points at all, hence certainly none that fail to lie in $A$.

For a little more general discussion of why mathematics deals with vacuous truth this way, you might also check out this answer..

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  • $\begingroup$ In general $\{ x\}$ is not closed. This is true for $X = \mathbb R$. Spaces $X$ in which all singletons are closed are called $T_1$-spaces. For example, if $X$ carries the trivial toploogy, then the closure of each singleton is the whole space $X$. $\endgroup$
    – Paul Frost
    Feb 4 at 17:59
  • $\begingroup$ @PaulFrost yes, OP had mentioned $\mathbb R$, but in any case I have edited to clarify. $\endgroup$
    – M W
    Feb 4 at 19:24

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