Page 189 - DMTH503_TOPOLOGY

Unit 20: The Urysohn Metrization Theorem

Notes
Example 4: A compact Hausdorff space is separable and metrizable if it is second countable.

Solution: Let (X, T) be a compact Hausdorff space which is second countable.
To prove that X is separable and metrizable.
Firstly, we shall show that X is regular.

X is a Hausdorff space.  X is a T -space.
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 X is also a T -space.
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 {x} is closed  x  X.
Let F  X be closed and x  X s.t. x  F.
Then F and {x} are disjoint closed subsets of X.

X is a compact Hausdorff space.
 X is a normal space.

As we know that “A compact Hausdorff space is normal”.
By definition of normality,
We can find a pair of open set G , G  X
1 2
s.t. {x}  G , F  G , G G = 
1 2 1 2
i.e. x  G , F  G , G G = 
1 2 1 2
 Given a closed set f and a point x  X s.t. x  F implies that  disjoint open sets G , G  X
1 2
s.t. x  G , F  G .
1 2
This implies X is a regular space. ...(2)
X is a second countable. [A second countable space is always separable ] ...(3)
 X is separable.
From (1), (2) and (3), it follows that (X, T) is a regular second countable T -space.
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And so by Urysohn’s theorem, it will follow that X is metrizable. ...(4)
From (3) and (4), it follows that X is separable and metrizable.

Hence the result.
Theorem 1: Every metrizable space is a normal Frechet space.
Proof: Let X be a metrizable space so that  a metric d on X which defines a topology T on X.

Step (i): To prove that (X, T) is a Frechet space i.e. T space.
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Let (X, d) be a metric space. Let x, y  X be arbitrary s.t. d(x, y) = 2r. Let T be a metric topology.

We know that every open sphere is T open. Then S , S are open sets s.t.
r(x) r(y)
x  S , y  S
r(x) r(x)
x  S , x  S
r(y) r(y)
Hence (x, d) is a T -space.
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