Page 9 - DMTH503_TOPOLOGY

Unit 1: Topological Spaces

1.1.2 Different Kinds of Topologies Notes

Stronger and Weaker Topologies

Let X be a set and let T and T be two topologies defined on X. If T  T , then T is called smaller
1 2 1 2 1
or weaker topology than T .
2
If T T , then we also say that T is longer or stronger topology than T .
1 2 2 1
Comparable and Non-comparable Topologies

Definitions: The topologies T and T are said to be comparable if T  T or T T .
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The topologies T and T are said to be non-comparable if T T and T T .
1 2 1 2 2 1

Example 3: If X = {s, t} then T = {, {s, X}} and T = {, {t}, X} are non-comparable as T  T
1 2 1 2
and T  T .
2 1
Discrete and Indiscrete Topology

Let X be any non-empty set and T be the collection of all subsets of X. Then T is called the discrete
topology on the set X. The topological space (X, T) is called a discrete space.
It may be noted that T in above definition satisfy the conditions of definition 1 and so is a
topology.
Let X be any non-empty set and T = {X, }. Then T is called the indiscrete topology and (X, T) is
said to be an indiscrete space.
Again, it may be checked that T satisfies the conditions of definition 1 and so is also a topology.

Example 4: If X = {a, b, c} and T is a topology on X with {a}  T, {b}  T, {c}  T, prove that
T is the discrete topology.

Solution: The subsets of X are:
X = , X = {a}, X = {b}, X = {c}, X = {a, b}, X = {a, c}, X = {b, c}, X = {a, b, c} = X
1 2 3 4 5 6 7 8
In order to prove that T is the discrete topology, we need to prove that each of these subsets
belongs to T. As T is a topology, so X and  belongs to T.

i.e. X  T, X  T.
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Clearly, X  T, X  T, X  T
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Now X = {a, b} = {a}  {b}
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since {a}  T, {b}  T (Given)

and T is a topology and so by definition 1, their union is also in T i.e. X = {a, b}  T
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similarly, X = {a, c} = {a}  {c}  T and X = {b, c} = {b}  {c}  T
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Hence, T is the discrete topology.

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