§ Quotient topology
I watched this for shits and giggles. I don't know enough topology at all, but it's
fun to watch arbitrary math videos.
§ Quotient topology: Defn, examples
Given space and equivalence relation on , the quotient set
inherits a topology. Let send point to
equivalence class. Quotient topology is the most refined topology
on such that is continuous. That is, it has the most open sets
for which this map is continuous.
- Intended to formalise identifications.
- More explicitly, a set (which is a collection of equivalence classes) is open iff is open in .
- Even more explicitly, is open iff is open in .
- Even more explicitly, we can write , because the elements of are equivalence classes.
§ Claim: quotient topology is a topology
- The preimage of the empty set is the empty set, and thus is open.
- The preimage of all equivalence classes is the full space, and thus open.
- Preimage of union is union of preimages: extend to get a new homotopy : and .
§ have HEP and is contractible, then
- Pick . We need another map such that their compositions are homotopic to the identities of and .
- Define as a section of , given by . This is a section of since (That is, maps entirely within the fibers of ).
- Consider . That is, lift from to using and then perform on . We claim that The map is the homotopy inverse of .
- (1a) is equal to , as , and .
- (1b) So we have , as is from defn, and is from homotopy. So we are done in this direction.
- (2a) Consider . We wish to show that this is continuous. Let's show that it lifs to a continous map upstairs. So consider . We claim that this is equal to , which is continuous as it is a composition of continuous maps.
- This relationship is hopefully intuitive: asks us to treat all of as if it were before applying . Since kills whatever does after, and guarantees to keep within , it's fine if we treat all of as just . asks us to treat as itself, and not . Since kills stuff anyway, we don't really care. The real crux of the argument is that where is a map that stabilizes .
- (2b) Consider --- Since , we crush all data regardless of what happens. This is the same as the value as and . For the other set, we get and hence we are done.
- (2c) Now since is continuous, and that , we are done since we can homotope from .
Slogan: Use HEP to find homotopy . Use as inverse to quotient.