Page 187 - DMTH503_TOPOLOGY

Unit 20: The Urysohn Metrization Theorem

d(x,y) Notes

write d (x, y) =  x, y X
1 1 d(x,y)

Then d is a metric on X.
1
d (x,y)

Again d (x, y) = 1  x, y X
2 1 d (x,y)

1
Then d is also a metric on X.
2
Continuing like this, we can define an infinite number of metrics on X.
Urysohn Metrization Theorem

Statement: Every regular second countable T -space is metrizable.
1
or

Every second countable normal space is metrizable.
Proof: Let (X, T) be regular second countable T -space.
1
To prove: (X, T) is metrizable.

X is regular and second countable.
 X is normal.
Since (X, T) is second countable and hence there exists countable base  for the topology T on X.
The elements of  can be enumerated as B , B , B ,..., where   B  T. Let x  X be arbitrary and
1 2 3 n
x  U  B.
By normality of X,

 V   s.t. x  U  V

Write C = {(U, V) : U × V   ×  s.t. V  V }
 is countable.   ×  is countable.

 every subset of  ×  is countable.
  is countable.
For    × 

U  V  U (X – V) = 

Also U and X – V are closed in the normal space (X, T).
Hence, by Urysohn’s lemma,

 continuous map f : X  [0, 1] = I, s.t.

f( U ) = {0}, f(X – V) = {1}

This implies f(x) = 0 iff x  U
and f(x) = 1 iff x  X – V

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