Page 150 - DMTH503_TOPOLOGY

Topology

Notes Take B = {A : n  N}.
x n
Evidently, B is a local base at x X for the usually topology on R.
x
Clearly, B ~ N under the map A n.
x n
Therefore, B is a countable local base at x X. But x X is arbitrary.
x
Hence R with usual topology is first countable.

16.1.2 Second Axiom of Countability

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.
In this case, the space X is called second countable or second axiom space.

Note A second countable space is also called completely separable space.

Example 2: The set of all open intervals (r, s) and r with s as rational numbers forms a
base, say B for the usual topology of R. Since Q, Q × Q are countable sets and so B is a countable
base for on R.

 (R, ) is second countable.

16.1.3 Hereditary Property

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.
E.g. first countable, second countable are hereditary properties, where as closed sets, open sets,
are not hereditary properties.
16.1.4 Theorems and Solved Examples on Countability Axioms

Theorem 1: Let (X, T) be a second axiom space and let C be any collection of disjoint open subsets
of X. Then C is a countable collection.

Proof: Let (X, T) be a second countable space, then a countable base
 = {B : n } for topology T on X.
n
Let C be a collection of disjoint open subsets of X.
Let A C be arbitrary, then A T.
By definition of base, B   st. B A.
n n
We associate with A, a least positive integer n s.t. B A.
n
Members of C are disjoint
 distinct integers will be associated with distinct member of C.
If we now order the members of C according to the order of associated integers, then we shall
get a sequence containing all the members of C. Hence, C is a countable collection.

Theorem 2: Let (X, T) be a first axiom space. Then is a nested (monotone decreasing) local base
at every point of X.

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