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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.
1     8
Clearly, X   T, X   T, X   T
2     3     4
Now X  = {a, b} = {a}  {b}
5
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
5
similarly, X  = {a, c} = {a}  {c}  T and X  = {b, c} = {b}  {c}  T
6                       7
Hence, T is the discrete topology.

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