Page 40 - DMTH503_TOPOLOGY

Topology

Notes A similar argument applies to  . Given a basis element (a, b) for T and a point x of (a, b), this
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same interval is a basis element for T that contains x. On the other hand, given the basis element
B = (–1, 1) –K for T and the point O of B, there is no open interval that contains O and lies in B.
Now, it can be easily shown that the topologies of  and  are not comparable.
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Self Assessment

9. Consider the following topologies on :
T = the standard topology,
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T = the topology of  ,
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T = the finite complement topology,
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T = the upper limit topology, having all sets (a, b) as basis,
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T = the topology having all sets (–, a) = {x : x < a} as basis
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Determine, for each of these topologies, which of the others it contains.

2.4 Summary

 A base (or basis)  for a topological space X with topology  is a collection of open sets in
T such that every open set in T can be written as a union of elements of .

 Sub-base: Let X be any set and  a collection of subsets of X. Then  is a sub-base if a base
of X can be formed by a finite intersection of elements of .
 Standard Topology: If  is the collection of all open intervals in the real line (a, b) = {x : a
< x < b}, the topology generated by  is called standard topology on the real line.
 Lower Limit Topology: If  is the collection of all half-open intervals of the form
[a, b) = {X : a  x < b},

where a< b, the topology generated by  is called the lower limit topology on .

2.5 Keywords

Finer: If T  T , then we say that T is longer or finer than T .
1 2 2 1
Subset: If A and B are sets and every element of A is also an element of B then, A is subset of B
denoted by A  B.
Topological Space: It is a set X together with T, a collection of subsets of X, satisfying the
following axioms.

(1) The empty set and X are in T.
(2) T is closed under arbitrary union.
(3) T is closed under finite intersection.

2.6 Review Questions

1. Let  be a basis for a topology on a non empty set X. It  is a collection of subsets of X such
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that T    , prove that  is also a basis for T.
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