From 28407333ffceca9b99fae721c30e8ae146a863da Mon Sep 17 00:00:00 2001 From: Justin Gassner Date: Wed, 14 Feb 2024 07:24:38 +0100 Subject: Update --- pages/general-topology/universal-constructions.md | 23 +++++++++++++++++++++++ 1 file changed, 23 insertions(+) create mode 100644 pages/general-topology/universal-constructions.md (limited to 'pages/general-topology/universal-constructions.md') diff --git a/pages/general-topology/universal-constructions.md b/pages/general-topology/universal-constructions.md new file mode 100644 index 0000000..827f730 --- /dev/null +++ b/pages/general-topology/universal-constructions.md @@ -0,0 +1,23 @@ +--- +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 %} -- cgit v1.2.3-54-g00ecf