Is the statement "A set is closed if and only if it contains all of its limit points" vacuously true for a singleton?

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?

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

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..

• 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 $X$. Feb 4 at 17:59
• @PaulFrost yes, OP had mentioned $\mathbb R$, but in any case I have edited to clarify.
– M W
Feb 4 at 19:24