# Gracious Living

Quotients of Topological Spaces
November 28, 2010, 20:06
Filed under: Math, Topology | Tags: , ,

The last way to induce a topological space is by taking a quotient.  This can be compared to taking a quotient of groups in the same way that the product topology corresponds to a direct product of groups, the subspace topology corresponds to a subgroup, and so on.  The quotient topology is a bit less intuitive than the other constructions we’ve done, but once you get the hang of it, it turns out to be very useful.

The simplest construction of the quotient is the categorical one.  We defined the subspace topology of $A\subset X$ to be the coarsest topology such that the inclusion map $i:A\hookrightarrow X$ was continuous.  Likewise, if we have a surjective map $q:X\rightarrow Z$, where $X$ is a topological space and $Z$ is a set, we define the quotient topology on $Z$ to be the finest such that $q$ is continuous.  This means that $U\subset Z$ is open if and only if $q^{-1}(U)\subset X$ is.  The “only if” part is the definition of continuity, but the “if” part defines what we’ll call a quotient map.  A quotient map then maps open sets in $X$ that are whole inverse images of sets in $Z$ to open sets in $Z$.  Open and closed maps both have this property, but there are quotient maps that aren’t open or closed.

In practice, you can look at all quotient maps in terms of equivalence relations.  Let $\sim$ be an equivalence relation on $X$, and let $X/\sim$ be the set of equivalence classes.  Let $q:X\rightarrow X/\sim$ send each point to its equivalence class.  This is a surjective map, and the quotient topology on $X/\sim$ has a set of equivalence classes in $X/\sim$ open if and only if their union is open in $X$.

One of the most common examples occurs when we have a closed subspace $A\subset X$.  Then an equivalence relation exists whereby all points in $A$ are equivalent, and each point in $X-A$ is in its own equivalence class.  Then $X/\sim=X-A\cup\{a\}$, where $a$ is the equivalence class of $A$.  A set is open in $X/\sim$ if and only if: it doesn’t contain $a$ and is open in $X-A$; or it contains $a$ and its preimage (given by replacing $a$ by $A$ again) is open in $X$.  Thus, $X/\sim$ is what we get when we shrink $A$ down to a single point.  We call this $X/A$.

You can also do this with $A$ open, but the resulting space has $\{a\}$ as an open set, and this is a Bad Thing.  When quotienting by an equivalence relation, you generally want equivalence classes to be closed for the same reason.

As an example, recall that $S^n=\{(x_1,\dotsc,x_{n+1})\in\mathbb{R}^{n+1}:x_1^2+\dotsb+x_{n+1}^2=1\}$ and $B^n=\{(x_1,\dotsc,x_n)\in\mathbb{R}^n:x_1^2+\dotsb+x_n^2\le 1\}$, so that $\partial B^n=S^{n-1}$.  We can start in dimension 1, and take the quotient of $B^1$, a line segment, by its boundary, $S^0=\{-1,1\}$.  $B^1/S^0$ is what we get when we glue the two endpoints of the line segment together, which is clearly homeomorphic to $S^1$.  Similarly, $B^n/S^{n-1}\approx S^n$ for each $n$.  This is one way of “inductively” thinking about higher dimensions.

Another example is the adjunction space of $f:A\rightarrow Y$, where $A\subset X$.  This is $X\cup_f Y=(X\sqcup Y)/\sim$, where $f(a)\sim a$ for all $a\in A$.  Adjunction is basically gluing $X$ to $Y$ along $A$, using $f$ to match $A$ with a subset of $Y$.  In the most common examples, $f$ is an embedding so we can regard $A$ as a subspace of either set.  For example, we can glue two disks together along their boundaries to get a copy of $S^2$.  On the other hand, we could instead have $g:A\rightarrow X$ instead of an inclusion map, so that $X\cup_{g,f} Y=(X\sqcup Y)/\sim$, where $f(a)\sim g(a)$ for $a\in A$.

A final important example is the partition into orbits of a group acting on a topological space.  For example, $\mathbb{Z}$ acts on $\mathbb{R}$ by translation, defining an equivalence relation $x\sim y$ if $x-y\in\mathbb{Z}$.  The quotient space $\mathbb{R}/\mathbb{Z}$ is then homeomorphic to $[0,1]/\{0,1\}=S^1$.  Of course, $\mathbb{R}$ is also an additive group with $\mathbb{Z}$ a normal subgroup, and the quotient group $\mathbb{R}/\mathbb{Z}$ is isomorphic to the circle group as a group.  This isn’t an accident — cosets of $\mathbb{Z}$, which are elements of the group $\mathbb{R}/\mathbb{Z}$, are precisely orbits under the action by $\mathbb{Z}$, which are points of the topological space $\mathbb{R}/\mathbb{Z}$.  Groups which also have a topological structure, like $\mathbb{R}$ or $S^1$, are quite important in mathematics.  (By the way, the notation $\mathbb{R}/\mathbb{Z}$ is highly ambiguous — we could also mean that we’re shrinking the subspace $\mathbb{Z}\subset\mathbb{R}$ down to a point, giving a countably infinite set of circles, all glued together at a single point.  Be sure you know what the author’s talking about when you see a quotient sign being used.)

Quotients preserve a couple properties: for example, a quotient of a connected space is still connected.  The most important property has to do with continuous maps.  If $Z$ and $X$ are topological spaces, $\sim$ is an equivalence relation partitioning $X$, and $g:X\rightarrow Z$ is a function that is constant on each equivalence class, then $g$ induces a unique map $f:X/\sim\rightarrow g$ such that $f\circ q=g$, where $q$ is the quotient map.  $f$ is continuous iff $g$ is.  So maps that are constant on equivalence classes factor through the quotient map corresponding to that equivalence relation.  The proof is really just applying definitions, so I leave it to you.

I went to Munkres to look stuff up for this post, but I was really disappointed by what I found.  Munkres is really concerned with the properties of a quotient map, which is a surjective map $q:X\rightarrow Y$ with $U\subset Y$ open iff $q^{-1}(U)\subset X$ is (so that $q$ induces the quotient topology on $Y$).  He proves, for instance, that the composite of two quotient maps is a quotient map, but that the product of two quotient maps is not a quotient map.  In my experiences with “applied” (algebraic/differential) topology, I’ve never come across a map that I needed to show was a quotient map, or a space that I needed to show was abstractly a quotient space.  Rather, the situation is invariably one where you have a space and a partition and you need to study properties of the quotient space, in which case you’re given a quotient map from almost the very start.  If you are interested in the properties of quotient maps and would like to see the above done a little more abstractly, please let me know, and I’ll go over these things in more detail.  Otherwise, I plan to move on to further subjects in topology.