Page 190 - DMTH503_TOPOLOGY

Topology

Notes Step (ii): To prove (X, T) is a normal space.
It follows by the theorem.
“Every metric space is normal space” proved in Unit -17.

Example 5: Every subspace of a metrizable is metrizable.
Solution: Let (Y, ) be a subspace of a metric space (X, d) which is metrizable so that

(i)  a topology T on X defined by the metric d on X.
(ii) Y  X and (x, y) = d(x, y)  x, y  X
Then the map  is a restriction of the map ‘d’ of Y. Consequently  defines the relative topology
 on Y, showing thereby Y is metrizable.

20.2 Summary

 Given any topological space (X, T), if it is possible to find a metric  on X which induces the
topology T then X is said to be the metrizable.

 The set R with usual topology is metrizable.
 Urysohn metrization theorem: Every second countable normal space is metrizable.
 Every metrizable space is a normal Frechet space.

20.3 Keywords

Compact: X is compact iff every open cover of X has a finite subcover.
Hausdorff : A topological space (X, T) is a Hausdorff space if given any two points x, y  X,  G,
H  T s.t. x  G, y  H, G H = .

Normal: Let X be a topological space where one-point sets are closed. Then X is normal if two
disjoint sets can be separated by open sets.
Regular: Let X be a topological space where one-point sets are closed. Then X is regular if a point
and a disjoint closed set can be separated by open sets.
T space: A topological space X is a T if given any two points x, y  X, x  y, there exists
1 1
neighbourhoods U of x such that y  Ux.
x
20.4 Review Questions

1. Give an example showing that a Hausdorff space with a countable basis need not be
metrizable.

2. Let X be a compact Hausdorff space. Show that X is metrizable if and only if X has a
countable basis.
3. Let X be a locally compact Hausdorff space. Let Y be the one-point compactification of X.
Is it true that if X has a countable basis, then Y is metrizable? Is it true that if Y is metrizable,
then X has a countable basis?

4. Let X be a compact Hausdorff space that is the union of the closed subspaces X and X .
1 2
If X and X are metrizable, show that X is metrizable.
1 2

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