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As promised, the quotient topology. I tried half-heartedly to get someone else in my class to write this post for me, since a) I don't like this topic and b) if I am honest, I'm not sure how much I understand it. But...trying to get someone to write a math blog post for me requires telling them about my math blog...and seeing as I already have enough of a nerdy rep in real life, I didn't need to augment that existing image of me.
In the words of my pretty awesome former GSI, "the quotient topology is just a way to glue topological space to make a new topological space". Here's a motivating example: consider the unit interval, [0, 1], as a topological space. Visually, this space is basically a line segment (oh yeah, I am a fourth-year math student and I make profound observations). What if there was a way to glue the endpoints together, so we get a circle? Is there a way to naturally derive a topology on this new set with 0 and 1 identified as the same point?
Well...so far, I have only asked one open-ended question on this blog, so it should be pretty obvious that the answer is YES. The topology is called the quotient topology, and it's kind of similar in a way to factor groups.
Usually, we form a quotient topology by building off of an equivalence relation, In the example of making a circle out of a line segment, we can define ~ on [0, 1] with 0~1 and all other elements as alone in their equivalence classes. Notationally speaking, [0, 1]/~ is the set of all equivalence classes of the equivalence relation ~. on [0, 1]. Then, the natural mapping q: [0, 1] → [0, 1]/~ defined by q(x) = [x] (where [x] is the equivalence class of x) is a quotient map. The quotient topology is the finest one such that q is still continuous; i.e., all sets with an open preimage are open. Then, [0, 1]/~ is the quotient space.
More generally, a quotient map q: X → Y is a surjective function that satisfies the condition: U ⊂ Y is open iff q-1(U) is open. The forward direction of this condition is the condition for continuity. The backwards direction, in normal words, says that if any preimage is open, then the original set it is a preimage of must also be open. Usually, quotient maps map from a set to the equivalence classes of the set under some equivalence relation.
Given a surjective map f: X → Y of sets, we can define a topology on Y so that it is the finest topology for which f continuous. This topology contains all the sets for which their preimage is open. (Notice why this topology is the finest: we can throw out some sets and the function would still be continuous since preimages of the remaining sets are open; but adding new sets will break the continuity of the function, since the preimages of these new sets will not be open.) This topology is the quotient topology, and Y is the quotient space.
Of course, all these ideas relate together. Endowing Y with the quotient topology will result in f being a quotient map. If q is a quotient map, then the topology on Y is a quotient space. In particular, if X is a topological space and ~ is some equivalence relation, q: X → X/~ is a surjective map so we can endow X/~ with the quotient topology so q is a quotient map.
I kind of relate this topic to factor groups/rings since you're mapping a set surjectively onto a partition of the original set; as a result, you get a new structure whose structure is somehow related to the structure of the original set. In this case, we map the set into onto equivalence classes, inducing a topology along the way. In algebra, we map a group onto cosets, inducing a group structure along the way.
Okay. Now that I've made good on my promise to write up quotient spaces, I can go back to fangirling over the marriage of algebra and topology in THE TOPOLOGICAL GROUP RETURNS: PART 2.
