Universal property of the quotient topology

Statement

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

The characteristic property of the quotient topology states that:

$f$ is continuous if and only if $f\circ p$ is continuous

[Expand]

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

---

---

TODO: classic suppose there's another style question

---

See also

• Quotient topology
• Passing to the quotient