Universal property of the quotient topology

Statement

[math]\require{AMScd} \begin{CD} (X,\mathcal{J}) @>p>> (Y,\mathcal{Q}_p)\\ @VVV @VVfV\\ \searrow @>>f\circ p> (Z,\mathcal{K}) \end{CD}[/math]

The characteristic property of the quotient topology states that:

[ilmath]f[/ilmath] is continuous if and only if [ilmath]f\circ p[/ilmath] is continuous

Proof that the quotient topology is the unique topology with this property

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TODO: classic suppose there's another style question

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See also

• Quotient topology
• Passing to the quotient