Page 235 - DMTH503_TOPOLOGY

Unit 28: Compactness in Metric Spaces

Notes
Write A = {(x : r  1,2,.....,K }.
n nr n
The set A can be constructed for each n  N.
n
A has the following properties:
n
(i) A is a finite set,
n
1
(ii) given x  X; x  A s.t. d(x, x )  .
nr n nr
n
Write A  A n
n N

Being a countable union of countable sets, A is enumerable

Clearly A  X
Taking closure of both sides, A  X  X i.e.

A  X [ X is closed in X]
We claim A = X

For this it is enough to show that X  A .
Let x  X be arbitrary and let G  X be an open set s.t. x  G.

By the property (ii) of A ,
n
1
Given, x  X,  x  A  A s.t. d(x , x) <  on taking  .  By the definition of open set in a
nr n nr
n
metric space.
X  G, G is open   positive real number r, S  G
(x,r)
 in particular S  G
(x, )
d(x, x ) <   x  S  G
nr nr (x, )
 x  G
nr
 G contains some points of A other than x.
 (G – {x}) A  

 x  D(A)  A

 x  A
Thus we have shown that

any x  X  x  A
This proves that X  A

Finally we have shown that
 A  X s.t. A is enumerable and A = X.

This proves that X is separable.

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