Page 153 - DMTH503_TOPOLOGY

Unit 16: The Countability Axioms

Proof: Let (Y, ) be a sub-space of a topological space (X, T) which is second countable so that Notes
there exists a countable base  for the topology T.
If we show that (Y, ) is second countable, the result will follow

 is countable   ~ N
  is expressible as
 = {B : N N}
n
Write

(i) Evidently  ~ N under the map Y B n.
1 n
  is countable.
1
(ii)  is a family of all –open sets.
1
For B   B T,
n n
B T  Y B 
n
(iii) any y  G B Y 
r 1
s.t. y Y B G.
r
For proving this let G 

s.t. y  G, then H  T s.t. G = H Y.
y G y H Y y  H and y Y.
By definition of base,

any y  H  T  B  s.t. y B H
r r
from which any y H Y 
 Y  B  s.t. y Y B G
r 1 r
i.e. any y  G   Y B  B
r 1
s.t. y Y B G.
r
Thus it follows that  is a countable base for the topology and Y. Consequently, (Y, ) is
1
second countable.
Theorem 7: A second countable space is always separable.
Proof: Let (X, T) be a second countable space.

To prove: (X, T) is separable.
Since X is second countable and hence a countable base  for the topology T on X. Members of
 may be enumerated as B , B , B , ... .
1 2 3
Choose an element x from each B and take A as the collection of all these x s.
i i i
That is to say, x  B   i  N ...(1)
c
i


and A {x : i N} ...(2)
i
Evidently N ~ A under the map i x
i
Therefore, A is enumerable.

LOVELY PROFESSIONAL UNIVERSITY 147