# Definition Given a [[Topological Space]] $(X, \mathcal{T})$, a family $\{A_i\}$ of [[subset|subsets]] of $X$ ($A_i \subset X \forall i \in I$) is called a *covering* of $X$ if $ \bigcup_{i \in I} A_i = X $ If $A_i \in \mathcal{T} \forall i \in I$ (i.e., they are [[open set|open sets]]), then the covering is said to be an open covering.