Page 157 - DMTH503_TOPOLOGY

Unit 16: The Countability Axioms

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(iii) Given any G  T with p G, r Q s.t. Notes
S G.
(p, r)
From what has been done, it follows that L is an enumerable local base at p of the topology
p
T on X.
16.2 Summary

 Let (X, T) be a topological space. The space X is said to satisfy the first axiom of countability
if X has a countable local base at each x X.
 Let (X, T) be a topological space. The space X is said to satisfy the second axiom of
countability if a countable base for T on X.

 Let (X, T) be a topological space. A property P of X is said to be hereditary if the property
is possessed by every subspace of X.

16.3 Keywords

Base:  is said to be a base for the topology T on X if x G T B  s.t. x BG.
Local Base: A family  of open subsets of X is said to be a local base at x X for the topology
x
T on X if
(i) any B   x B
x
(ii) any G T with y G B  s.t. y B G.
x
+
Open Sphere: Let (X, ) be a metric space. Let x X and r R . Then set {x X :  (x , x) < r} is
0 0
defined as open sphere with centre x and radius r.
0
Separable: Let X be a topological space and A be a subset of X, then X is said to be separable if
(i) A  X

(ii) A is countable.
16.4 Review Questions

1. Prove that the property of being a first axiom space is a topological property.
2. For each point x in a first axiom T – space,
1
{x}  n N B (x)
n

3. Prove that the property of being a second axiom space is a topological property.
4. In a second axiom T – space, a set is compact iff it is countable compact.
1
5. Show that in a second axiom space, every collection of non empty disjoint open sets is
countable.
6. Give an example of a separable space which is not second countable.
7. Show that every separable metric space is second countable. Is a separable topological
space is second countable? Justify your answer.

8. Every sub-space of a second countable space is second countable and hence show that it is
also separable.

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