# Definition Consider a [[Topological Space]] $(X, \mathcal{T})$ all possible [[Covering of a Set|coverings]] of $X$. $X$ is said to be *compact* if for every [[Open Set|open]] covering $\{U_i|i\in I\}$, there exists a *finite* [[subset|subset]] $J \subset I$ such that $\{U_j|j \in J\}$ is also a covering of $X$. The notion of compactness generalizes the notion of closedness and boundedness from Euclidean space to topological spaces. In fact, it is a theorem that if $X \subset \mathbb{R}^n$, then $X$ is compact if and only if it is [[Closed Set|closed]] and bounded.