Page 156 - DMTH503_TOPOLOGY

Topology

Notes Finally, any y  S (x,r)  y  S (x, )

 S  S
(a, r) (x, )
From (2) and (3), (a,x) r, so that x   S (a,r) .

Thus, we have shown that

x S (a,r)  S (x, )  G.

from which x S (a,r)  G.


Thus, x G T    Q s.t. x  S (a,r)

r

T

i.e. x G    S (a,r)  s.t. x  S (a,r)  G.
This proves that  is a base for the topology T on X. From what has been done, it follows
that  is enumerable base for the topology T on X and hence X is second countable.
Example 5: Every separable metric space is second countable.
Solution: Refer second part of the above theorem.

Theorem 10: A metric space is first countable.
Proof: Let (X, ) be a metric space. Let T be metric topology on X, corresponding to the metric 
on X. Let X be arbitrary.
To prove that (X, T) is first countable, it suffices to show that a countable local base at p for the
topology T on X.
+
Write L = {S : r  Q }.
p (p, r)
Q is enumerable and hence its subset Q ,
+
+
Q is enumerable L is enumerable.
p
Let G T be arbitrary s.t. G.
Then, by definition of an open set.
s R s.t. S G.
+
(p, s)
Choose a positive rational number r s.t. r < s.
Then S (p,r)  S (p,s)  G

or S (p,r)  G .

Given any G T with  G.
r Q s.t. S G.
+
(p, r)
Now L has the following properties:
p
(i) every member of L is an open set containing p.
p
each open sphere forms an open set.

(ii) L is enumerable set.
p

150 LOVELY PROFESSIONAL UNIVERSITY