Product space revisit
Let represent the cartesian product (笛卡尔积) of
Both 1 and 2, if is a subspace of , is a subspace of .
Both 1 and 2, if each space is Hausdorff, then is Hausdorff.
Both 1 and 2, if ,then
Only 2. Let be given by ,while called the coordinate function.Then is cont. iff. is cont.,
When 1, try to find contradiction by the fact that infinite intersection of open sets may not open. (J -> inf）
A surjective map is said to be a quotient map if
a subset is open in Y iff. is open in X.
each open set maps to an open set
each closed set maps to an closed set
If a surjective cont. map p is open or closed, p is a quotient map.
Bijective quotient map is a homeomorphism
A is a subset of X, then a retraction of X onto A is a cont. map s.t. r(a)=a, it's a quotient map.
composition（复合） of quotient maps is quotient
If is a surjection, then there is exactly one topology to make p a quotient map.It's called the quotient topology induced by p.
Just make sure that
Let be a partition(分划) of X,p is the cannonical projection
The quotient topology induced by p on is called a quotient space