is called the standard bounded metric on R
,we can define as a metric on which gives the product topology on
Let X be a space with T(1) condition
suppose is a family of continuous maps satisfying that for all x0 in X and each open neighborhood U of x0, there is an index n so that fn is positive at x0 and zero outside U
Then is an inbedding of X in
Let (X,d) be a metric space, (X,Td) is the corresponding topology space
1) A, a subset of X, as a topology subspace of (X,Td)
2) A with metric
1) is same as 2)
Tietze extension theorem
Let X be a normal space, let A be a closed subspace of X, then we have:
(1) Any continuous may be extended to a continuous map so that
(2) Any continuous can be extended to a continuous map so that
Every regular space with a countable basis is normal
Urysohn metrization theorem
metrizable : there is a metric d which can generate (X,T) by Td.
Every regular space with countable basis is metrizable
there exists a countable collection of continuous functions having the property that :
Given any point and any neighbourhood U of , there exists an index n so that fn is positive at and zero outside U.
partition of unity
Let be a finite indexed open cover of X
An indexed family of continuous maps is called a partition of unity dominated by if:
Existence of finite partition of unity
any open cover of a normal space X can dominate a partition of unity.