Page 155 - DMTH503_TOPOLOGY

Unit 16: The Countability Axioms

16.1.5 Theorems Related to Metric Spaces Notes

Theorem 9: A metric space is second countable iff it is separable.
Proof:
(i) Let (x, ) be a metric space. Let T be the metric topology on X corresponding to the metric
. Let (X, T) be second countable. To prove that X is separable.

Here write the complete proof of the theorem (6).
(ii) Conversely, suppose that (x, ) is a metric space and T is a metric topology on X
corresponding to the metric . Also, suppose that X is separable, so that

 A  X s.t. A  X and A is countable.

A is countable A is expressible as
A = {a : n N}
n
To prove that X is second countable.
We know that each open sphere forms an open set.
Let a A be arbitrary.
n
+
Write  = {S (a ) : r Q , n  N}.
r n
Q is an enumerable set
+
 Q is an enumerable set
 Q Q
+
Then  is a countable base for the topology T on X.
 X is second countable.

Let G  T be arbitrary s.t. x G.
x being an arbitrary point of X.
By definition of open set in a metric space,
 a positive real number  s.t. S G ...(1)
(x, )
Since A is dense in X and so there will exist a point a A s.t.


 (a,x)  ...(2)
3
Since Q is dense in R for the usual topology on R and hence its subset Q is also dense in R
+
+
with usual topology so that r  Q s.t.
 2 
 r 
3 3
Aim: S r(a)  S  (x)  G.
Also let y S (a,r) be arbitrary so that (y,a) r  ...(3)

 (x,y)   (x,a)   (a,y)
  2 
<  r   , from (3)
3 3 3
 (x,y)   y S (x, )


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