A property of quotient maps

general-topology

Wikipedia says:

Quotient maps $q:X\to Y$ are characterized among surjective maps by the following property: if $Z$ is any topological space and $f:Y\to Z$ is any function, then $f$ is continuous if and only if $f\circ q$ is continuous.

What does it mean that they are characterized among surjective maps?

Does it mean the following: suppose $q:X\to Y$ is a surjective map, and $Y$ is equipped with a topology that has the property that if $Z$ is any topological space and $f:Y\to Z$ is any function, then $f$ is continuous if and only if $f\circ q$ is continuous. Then it follows that the topology on $Y$ is the final topology with respect to $q$.

But I don't see why this should be true.

Also, my topology lecturer called the property from the Wikipedia quote above a "universal property" of quotient spaces. I think this terminology is not quite accurate, since it is not a universal property in the sense of category theory. Can you confirm that thought?

Best Answer

For a given function $q:X\to Y$ lets define the universal property to mean: for any $f:Y\to Z$ we have that $f$ is continuous if and only if $f\circ q$ is.

All we need to show is that if $q:X\to Y$ is surjective and satisfies the universal property then $q$ is a quotient map.

Proof. So assume that $q$ is not a quotient map. Which means that at least one of the following holds:

  1. There is a subset $U\subseteq Y$ open but $q^{-1}(U)$ is not open. Or in other words $q$ is not continuous. Then we can apply our universal property to the identity $id:Y\to Y$, $id(x)=x$ which is continuous, to obtain contradiction.
  2. There is a subset $U\subseteq Y$ which is not open but $q^{-1}(U)$ is. Now consider $Z=\{0,1\}$ with the Sierpiński topology, i.e. $\{1\}$ is open but $\{0\}$ is not. Finally let $f:Y\to Z$ be given by $$f(x)=\begin{cases} 1&\text{if }x\in U\\ 0&\text{otherwise} \end{cases}$$ Clearly $f$ is not continuous. However $f\circ q$ is continuous. Contradicting our universal property on $q$. $\Box$

Of course there's another implication to prove here: if $q$ is a quotient map then it satisfies the universal property. Which I leave as an exercise.

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