Suppose \(B\) is a set and \(S = \{ f_{\alpha} X_{\alpha} \to B \}\) is a family of functions to topological spaces. The final topology on \(B\) determined by \(S\) is the unique topology on \(B\) with the property: A function \(F:B \to Z\) to a space \(Z\) is continuous if and only if \(F \circ f_{\alpha}\) is continuous for every \(\alpha\).