# Definition Given a [[topological space]] $(X, \mathcal{T})$ endowed with a [[Metric]] $d$. This metric makes the [[open set|open sets]] of $X$ into open discs: $ U_{\varepsilon}(X) = \{y \in X | d(x,y) < \varepsilon\}. $ The metric topology determined by $d$ is $\mathcal{T}$ that contains all $U_{\varepsilon}$ for all $\varepsilon$ and all their possible unions.