For example, consider the set: $$\{2,\ 3,\ 5\} \subset \mathbb R.$$
This set has no limit points.
A closed set (also this) is a set which contains all of its limit points.
The set described above contains all of its $0$ limit points, therefore it is closed.
Is this reasoning correct? Can it be made rigorous more other than just re-writing it with quantifiers?
Thank you.
Best Answer
Right, the set is closed, for the exact reason you described.
$$\forall x\in \emptyset: P(x)$$ is vacuously true regardless of $P(x)$. This is perfectly rigorous reasoning (in the context of real analysis).