The subspace topology
If Y is a subset of X, then is a topology on Y, called the subspce topology. is called a subspace of .
If is a basis for , then is a basis for the subspace topology on Y. （交并分配律）
If A is a subspace of X, B is a subspace of Y, then the product topology on A×B inherits as a subspace of X×Y.
Topology and Basis
(2) The union of any collection of elements of T is in T （开集之并必开）
(3) The intersection of any finite collection of elements of T is in T
(X,T) is called a topological space
,by convention,call u an open
defines a topology on X
A basis is not unique for a given topology
equals the collections of all unions of elements of
（“开”由 定义 开集之集就是任意些B的并集之集）
Suppose is a collection of open sets of X,then is a basis for T if
a collection of subsets of X, whose union is X（X的某个覆盖）
topology generated by S is defined to be the collection of all unions of finite intersections of elements in S
Collection of all finite intersection of elements in S is a basis
If call A a closed set
(1) ∅，X are closed （空集、全集必闭）
(2) Arbitrary intersection of closed sets is closed（闭集之交必闭）
(3) Finite unions of closed sets are closed（闭集之有限并必闭）
A is closed in ,C is closed in X
If A is closed in Y, and Y is closed in X, then A is closed in X.
The product topology on
the topology having as basis
If and are basis of respectively, then is a basis for the product topology.（B们可精准覆盖的=B'们可精准覆盖的）
is a subbasis for the products topology
Closure and interior of a set
The interior of A( )
The closure of A( )
the intersection of all closed sets in X contianing A.
If A is open, A=
If A is closed, A=