Page 9 - DMTH503_TOPOLOGY

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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 .
1 2 1 2 2 1
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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