Main Question or Discussion Point
Does anyone have any good reference to exercises concerning these topics? I would like to understand them better. Thank you.
I understand that, I'm just a bit unsure of why the quotient topology is the "right" one. Is it just because then these spaces are homeomorphic to each other (like you said, identifying the endpoints of an interval in the quotient space makes it homeomorphic to the circle in the quotient topology)? Or is there something deeper and more subtle I am missing.the point is to understand complicated spaces in terms of simpler ones.
i.e. an interval is simpler than a circle but a circle i q quotient o an interval.
'a torus is moire complicated than a rectangle but a torus is a quotient of a rectangle,....
almost any space is a successive union of quotients of rectangles of various dimensions.