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+---
+title: Universal Constructions
+parent: General Topology
+nav_order: 1
+---
+
+# {{ page.title }}
+
+{% definition Initial Topology %}
+Suppose that $f_i : S \to X_i$, $i \in I$, is a family of maps,
+from a set $S$ into topological spaces $X_i$.
+The *initial topology* on $S$ induced by the family $(f_i)$
+is defined to be the weakest topology on $S$
+making all maps $f_i$ continuous.
+{% enddefinition %}
+
+{% theorem * Universal Property of the Initial Topology %}
+The initial topology on $S$ induced by the family $(f_i)$
+is the unique topology on $S$ with the property that
+for any topological space $T$,
+a mapping $g : T \to S$ is continuous if and only if
+all compositions $f_i \circ g : T \to X_i$ are continuous.
+{% endtheorem %}