Notes:Universal property of the quotient topology

Problem

I really want to do this using equivalence relations rather than a mapping, I also need to work out exactly which theorem is which.

Theorems

John M. Lee

Characteristic property of the quotient topology

$\xymatrix{ X \ar[d]_q \ar@{->}[dr]^{f\circ q} & \\ Y \ar[r]_f & Z}$
Diagram

• Suppose $(X,\mathcal{ J })$ and $(Y,\mathcal{ K })$ are topological spaces
• Suppose $q:X\rightarrow Y$ is a quotient map, then:
• For any topological space $(Z,\mathcal{ L })$ and any map $f:X\rightarrow Z$
• $f$ is continuous $\iff$ $f\circ q$ is continuous

Passing to the quotient

$\xymatrix{ X \ar[d]_q \ar[dr]^{f} & \\ Y \ar@{.>}[r]_{\bar{f} } & Z}$
Diagram

• Suppose $(X,\mathcal{ J })$ and $(Y,\mathcal{ K })$ are topological spaces
• Suppose $q:X\rightarrow Y$ is a quotient map, then:
• Suppose $(Z,\mathcal{ L })$ is a topological space and $f:X\rightarrow Z$ is any map that is constant on the fibres of $q$
• Then there exists a unique continuous map $\bar{f}:Y\rightarrow Z$ such that $f=\bar{f}\circ q$

Munkres

Theorem 22.2

This is just "passing to the quotient" but stated slightly differently.

Theory

Lee's characteristic property does say "you can stick any topology you like, if there's a continuous map from $\frac{X}{\sim}$ to it, then there's a continuous map from the parent to it" which is "universal"

Passing to the quotient uses that and is something else. Very close concept, but still distinct.