Page 36 - DMTH503_TOPOLOGY

Topology

Notes Another Definition:
 is said to be a base for the topology T on X if x  G  T   B   s.t. x  B  G.
The elements of  are referred to as basic open sets.

Example 1:
(1) , the standard topology on , is generated by the basis of open intervals (a,b) where
a < b.
(2) A basis for another topology on  is given by half open intervals [a,b), a < b. It generated
the lower limit topology .
(3) The Open intervals (a,b), a < b with a & b rational is a countable basis. It generates the same
topology as .

Example 2: Let X = {1, 2, 3, 4}. Let A = {{1, 2}, {2, 4}, {3}}. Determine the topology on X
generated by the elements of A and hence determine the base for this topology.

Solution:
Let X = {1, 2, 3, 4} and
A = {{1, 2}, {2, 4}, {3}}.

Finite intersections of the members of A form the class  given by
 = {{1, 2}, {3}, {2, 4}, , {2}, X}.

The unions of the members of  form the class T given by
T = {{1, 2}, {3}, {2, 4}, , {2}, X, {1, 2, 3}, {1, 2, 4}, {3, 2, 4}, {3, 2}}.
It can be easily verified that  is a base for the topology T on X.

2.1.1 Topology Generated by Basis

Lemma 1: Let  be a basis for a topology T on a set X. Then T equals the collection of all unions
of elements of .

Proof: Each element of  is open, so arbitrary unions of elements in  are open i.e., in T. We
must show any  T equals a union of basis elements. For each x  , choose a set   that
x
contains x.
What does the union  of these basis elements equal? All of i.e. a union of basis
x x is
elements. How to find a basis for your topology.
Lemma 2: Let (X, T) be a topological space. Suppose  is a collection of open sets of X s.t.  open
sets and  x  , there exists an element B   s.t. x  B  . Then  is a basis for T.
Proof: We show the two basis conditions:
1. Since X itself is open in the topology, our hypothesis tells us that  x  X, there exists B  
containing x.
2. Let x  B  B . Since B , B are open, so is B  B ; by our hypothesis, there exists B  
1 2 1 2 1 2
containing x with B  B  B .
1 2

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