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In topology class, I often hear statements like "It turns out that $f$ is always a quotient map." I don't understand what is so special about quotients that the question of whether a given surjective map is or isn't a quotient is so interesting. What motivates these statements, and what should I think about when someone mentions that $f$ is a quotient?

lily
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    I'm no topologist, but when I hear "quotient" in a topological context, I think of drops of glue applied to selected points of the space. Like what you do to turn a rectangular piece of paper into a Möbius band, or the less curiour regular band, or even a doughnut surface (or was it a coffee mug). Of course, there are pathological possibilities like applying glue to a dense subset and such (think $\Bbb{R}/\Bbb{Q}$). If you encounter something like that my advice is not to sniff the glue. – Jyrki Lahtonen Feb 28 '18 at 07:03
  • @JyrkiLahtonen Thanks for the thoughts. My question is less about the quotient construction itself, and more about the situation where I have a map to begin with, and then I find out that it is a quotient map. Just to add some context to the example you mentioned, there's a discussion of $\mathbb{R}/\mathbb{Q}$ here. – lily Feb 28 '18 at 07:37
  • I can't see what kind of answer you could possibly want besides the obvious one: one cares about quotient maps because one cares about quotients. Would you ask the analogous question for surjective homomorpisms in group theory? Surjections per se don't matter in the presence of extra algebraic or topological structure. Surjections that respect the extra structure do matter. – symplectomorphic Feb 28 '18 at 07:54
  • @symplectomorphic I was already assuming that a "map" is continuous of course. But it is not the case that all surjective maps are quotient maps, so I am specifically asking what is special about quotient maps. Also, your "answer" is not so helpful - Why do we care about quotients, then? – lily Mar 05 '18 at 09:25

2 Answers2

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Quotient maps preserve more properties of spaces than mere continuous maps do. E.g. we know that if $f:X \to Y$ is surjective and continuous and $X$ is connected then so is $Y$. The same does not hold for locally connected $X$ in general, but it does hold when $f$ has the stronger property of being a quotient map.

It also helps that other classes of maps (open surjective maps, closed surjective maps and perfect maps) are special cases of quotient maps.

Some properties are chacterised by quotient maps: A space is a $k$-space iff it is a quotient image of a locally compact Hausdorff space and a space is sequential iff it is the quotient image of a metrisable space, e.g. It implies that both these properties are preserved by quotient maps as well.

Another handy fact: if $f: X \to Y$ is quotient we can test the continuity of a function $g: Y \to Z$ by seeing whether $g \circ f: X \to Z$ is continuous. This universal property of quotients is important in category theory.

Henno Brandsma
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Quotient maps are fun once you get the hang of them.
A quotient space is a partition of a space S into sets,
each of which is considered to be a point in the quotient
space. The quotient map f, is a map from each point x, of S
to the partition part in which x resides.

The topology of the quotient space is created by requiring
f to be continuous.

In group theory, a simular partioning occures and the
quotient is required to be a group. That requires a
normal subgroup to be the identity of the quotient group.

They also can be lots of fun giving us,
for example, the integers modulus n.

William Elliot
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