Trouble understanding quotient spaces

Question

I’m trying to grasp the idea behind quotient spaces and reading this wikipedia article. In the section ”Examples” they have the unit square $S^2$ homeomorphism example, which I tought would be something I could use to start building my understading of these. I’ve gone through abstract algebra course and now what an equivalence relation is however I still cannot understand the idea here.

Consider the unit square $I^2 = [0,1] × [0,1]$ and the equivalence relation $\sim$ generated by the requirement that all boundary points be equivalent, thus identifying all boundary points to a single equivalence class. Then $I^2/ \sim$ is homeomorphic to the sphere $S^2$.

The problem is with the sentence.

equivalence relation $\sim$ generated by the requirement that all boundary points be equivalent, thus identifying all boundary points to a single equivalence class.

What does this exactly mean? All the boundary points of $I^2$ are the set $\partial I^2$ which I guess could be denoted as $[0,1] \times \{0\} \cup \{0\} \times [0,1] \cup [0,1] \times \{1\} \cup \{1\} \times [0,1]$? Also the sentence ”thus identifying all boundary points to a single equivalence class.” is somewhat confusing. Any clarification for this would be greatly appreciated.

Answer 1

The equivalence relation $\sim$ is this one:

• if $p\in I^2\setminus\partial I^2$, then $p\sim q$ if and only if $q=p$;
• if $p\in\partial I^2$, then you have $p\sim q$ if and only if $q\in\partial I^2$ too.

So, there are two types of equivalence classes:

• those which consist of a single point from $I^2\setminus\partial I^2$;
• an equivalence class equal to $\partial I^2$.

So, the set $I^2/{\sim}$, which is the set of all equivalence classes, is the set$$\bigl\{\{p\}\mid p\in I^2\setminus\partial I^2\}\cup\bigl\{\partial I^2\bigr\}.$$The essential idea here is that, in $I^2/{\sim}$, the whole boundary of $I^2$ consists now of a single point.

Answer 2

Let A be the subset consisting of boundary points. Then define an equivalence relation on $I^2$ by declaring $x\sim y$ if and only if $x=y$ or $x,y \in A$ then the resulting quotient space is the space youre talking about and you can geometrically see is $S^2$.

The whole concept of quotient space is constructed so that you can treat points not equal in the original space as equal in the quotient space (identifying points)