Lebesgue's Number Lemma

Definition

Consider a Metric Space $(X,\mathcal{T},d)$, and an open covering $\{U_i{\}}_{i\in I}\subset \mathcal{T}$. If $X$ is compact, then $\exists \delta >0\text{s.t.}\forall B\subset X\text{s.t.}\mathrm{diam}(B)<\delta ,\exists U_i\in \{U_i{\}}_{i\in I}\text{s.t.}{U}_i\supset B$ where $\delta$ is called a Lebesgue number for the covering $\{U_i{\}}_{i\in I}$.