What does the quotient space of a CW complex by a subcomplex mean?

Question

From p.8 of Hatcher's Algebraic Topology:

If $(X,A)$ is a CW pair consisting of a cell complex $X$ and a subcomplex $A$, then the quotient space $X/A$ inherits a natural cell complex structure from $X$.

It talks about a quotient space of $X/A$ where $A$ is a subcomplex of $X$. What is the equivalence relation forming this quotient space? Thanks.

Somehow Zev Chonoles got that to work when he edited my other post. I guess it's a moderator privelege to auto-generate a screenshot from a given link. — Daniel Donnelly, Aug 03 '13 at 22:55
@EnjoysMath: I did not auto-generate any screenshot. Just take a screenshot as you would normally from your computer of the relevant piece of the document. Then upload it as a picture (which any user with reputation of $\geq 10$ can do). By the way, I'm not a moderator anymore. — Zev Chonoles, Aug 03 '13 at 23:02

score 2 · Accepted Answer · answered Aug 03 '13 at 22:57

In general, if $X$ is a topological space, and $A$ is a subspace, the notation $X/A$ refers to the space obtained by identifying all of $A$, ie, collapsing $A$ to a point.

Explicitly, the equivalence relation is $x\sim y$ if and only if $x,y\in A$. Notice this leaves all points in $X\setminus A$ unidentified. The topology of $X/A$ is just the quotient topology under $\sim$.

As an example, let $X=S^2$ and $A$ be an equator. Then $X/A$ pinches the equator to a point, leaving the wedge product of two spheres.

What does the quotient space of a CW complex by a subcomplex mean?

1 Answers1