Proposition

Given a Topological Space, $(X,\mathscr{T})$, the following statements are equivalent:

1. $A\subseteq X$ is Closed.
2. $A^{c}=X\setminus A$ is Open.
3. $A=\bar{A}$, i.e. $A$ is equal to its Closure.
4. $A$ contains all its limit points, i.e. $$\forall (x_{n})_{n\in \mathbb{N}}\subset A \text{ s.t. } \exists x\in X:x_{n}\to x \implies x\in A.$$
5. $A$ contains all of its boundary points