A Set is Closed iff it is its own Closure

Last updated Nov 1, 2022

Let $(X, \tau)$ be a Topological Space. Then $K \subset X$ is Closed If and Only If $\text{cl} K = K$.

($\Rightarrow$) Suppose $K$ is Closed. A Closure is defined as

$$\text{cl} K = \bigcap\limits {R : R \text{ is closed in } X, R \supset K}$$

$K$ is one such Set in the intersection so $\text{cl} K \subset K$. However, all sets in the intersection contain $K$ so $\text{cl} K \supset K$. Therefore $\text{cl} K = K$. $\checkmark$

($\Leftarrow$) Suppose $\text{cl} K = K$. Since $\text{cl} K$ is Closed, we have that $K$ is Closed. $\checkmark$ $\blacksquare$