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Let \(A\) be a set and \(\{F_{i}: X_{i} \to A\}_{i \in I}\) a family
of functions to topological spaces. There is a unique finest
topology on \(A\), relative to which every \(F_{\i}\) is
continuous. This is the final topology.
Take a set \(A\) and \(S = \{ f_{\alpha}: A \to X_{\alpha} \}\) a set of
functions to topological spaces. The initial topology on \(A\)
determined by \(S\) is the unique 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 \(\alpha\).