# Trouble understanding quotient spaces

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.

• It means that you can treat all points in the boundary as one - if this confuses you, get a piece of paper, and try to glue all boundary points together. It should look like a 2-sphere after gluing. Commented Jul 20, 2021 at 11:38

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.

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)