Let \(A\) be a set and \(\{F_{i}:A \to X_{i}\}_{i \in I}\) a family of functions to topological spaces. There is a unique coarsest topology on \(A\), relative to which every \(F_{i}\) is continuous. This is the initial topology denoted \(\mathcal{T}_{i}\). That is, if there is another topology \(\mathcal{T}\) for which \(F_{i}\) is continuous for all \(i \in I\), then \(\mathcal{T}_{i} \subseteq \mathcal{T}\).