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?
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..
