Page 151 - DMTH503_TOPOLOGY

Unit 16: The Countability Axioms

Proof: Let (X, T) be first axiom space, then is a countable local base Notes
B (x) = {B : n N} at every point x X.
n
Write C = B , B = B  B , C = B B B , ....,
1 1 2 1 2 3 1 2 3
n
C = B .
i
n

i 1
Then C C C ... C .
1 2 3 n
x B   n  x C  T  n.


n
n
It follows that C(x) = {C : n N} is a nested local base at x.
n
Theorem 3: A second countable space is always first countable space.
Or
Prove that second axiom of countability first axiom of countability.
Proof: Let (X, T) be a topological space which satisfies the second axiom of countability so that
(X, T) is second countable.
To prove that (X, T) also satisfies the first axiom of countability.
i.e., to prove that (X, T) is first countable.

By hypothesis, a countable base  for topology T on X.
 is countable  ~ N
This show that  can be expressed as
 = {B : n N}
n
Let x X be arbitrary.

Write L = {B x B }
x n n
(i) L , being a subset of a countable set , is countable.
x
(ii) Since members of  are T open sets and hence the members of L . For L .
x x
(iii) Any G L x G, according to the construction of L .
x x
(iv) Let G T for arbitrary s.t. x G.
Then, by definition of base,
x  G J  B  s.t. x  B G,
r r
 B L s.t. B G,
r x r
For B  with x B B L .
r r r x
Finally x G T  B L s.t. B G. ...(1)
r x r
From (i), (ii), (iii), (iv) and (1), it follows that L is a countable local base at x X. Hence, by
x
definition, X is first countable.
Theorem 4: To prove that first countable space does not imply second countable space. Give an
example of a first countable space which does not imply second countable space.

Proof: We need only give an example of a space which does satisfy the first axiom of countability
but not the second axiom of countability.

LOVELY PROFESSIONAL UNIVERSITY 145