pages/general-topology/universal-constructions.md


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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 %}