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, is Hausdorff.
Both 1 and 2, if ,then
Only 2. Let be given by ,while is called the coordinate function.Then is continuous iff. is continuous, .
When 1, try to find contradiction by
A surjective map is said to be a quotient map if
a subset is open in Y iff. is open in X.
a quotient map is sure to be continuous
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(正则映射)
The quotient topology induced by p on is called a quotient space