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.