Closure
# Definition
Let $X, \tau$ be a Topological Space and let $S \subset X$. Then the Closure of $S$, denoted $\text{cl}S$ is
$$\text{cl}S := \bigcap\limits {K \subset X : K \supset S, K \text{ is closed in } X}$$
# Properties
$\text{cl} S$ is Closed in $X$ since an Intersection of Closed Sets is Closed
$\text{cl}S = \bar{S} := \bigcup\limits {x \in X : x \text{ is a limit point of } S}$. That is, Closure of a Set is all its Limit Points.
$\text{cl}S = X \setminus \text{int} S^{C}$