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 \(A\) determined by \(S\) is the unique topology on \(A\) with the property that A function \(F:Z \to A\) from a topological space \(Z\) is continuous if and only if \(f_{\alpha} \circ F\) is continuous for every \(\alpha\). Note: The clockwise direction is just composition of continuous functions. The power comes from \(f_{\alpha} \circ F\) continuous \(\implies F\) continuous.

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 \(\alpha\).