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