Difference between revisions of "Universal property of the quotient topology"

Latest revision as of 22:17, 9 October 2016

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-==Statement==
+#REDIRECT [[Characteristic property of the quotient topology]]
-<math>\require{AMScd}
+{{Theorem Of|Topology}}
-\begin{CD}
-(X,\mathcal{J}) @&gt;p&gt;&gt; (Y,\mathcal{Q}_p)\\
-@VVV @VVfV\\
-\searrow @&gt;&gt;f\circ p&gt; (Z,\mathcal{K})
-\end{CD}</math>
-The characteristic property of the [[Quotient topology|quotient topology]] states that<ref>Introduction to topological manifolds - John M Lee - Second edition</ref>:
-{{M|f}} is continuous '''if and only if''' {{M|f\circ p}} is continuous
-{{Begin Theorem}}
-Proof that the quotient topology is the unique topology with this property
-{{Begin Proof}}
-{{Todo|classic suppose there's another style question}}
-{{End Proof}}
-{{End Theorem}}
-==See also==
-* [[Quotient topology]]
-* [[Passing to the quotient]]
-==References==
-<references/>
-{{Theorem|Topology}}