Introduction to Topological Spaces and Examples

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Prof.Retheesh R
Department of Mathematics
 
INTRODUCTION
TOPOLOGICAL SPACES
EXAMPLES OF TOPOLOGICAL
SPACES
BASES AND SUB BASES
REFERENCES
 
 
 
 
The word 
Topology
 is derived from the two
Greek words 
topos
 meaning ‘surface’ and
logos
 meaning ‘discourse’ or ‘study’.
 
Topology thus literally means study of
surfaces.
 
Definitions
 
Open ball: Let 
x
0
ϵX and r be a positive real number.
Then the open ball with centre x
0
 and radius r is defined to
be the set { xϵ X: d(x, x
0
)<r } which is denoted either
by B
r
(x
0
) or by B(x
0
,r). It is also called open r ball
around x
0
.
Open set: A subset A 
Ϲ
 
X is said to be open if for
every 
x
0 
ϵ 
A there exists some open ball around
x
0
  which is contained in A, that is ,there exists
r>0 such that 
B(x
0
,r)
 
Ϲ
 
A.
 
TOPOLOGICAL SPACE
 
A topological space is a pair (X ,Ʈ) where X is
a set and Ʈ is a family of subsets of X
satisfying.
i.
ɸ 
ϵ
 Ʈ and X
 ϵ
 Ʈ
ii.
 Ʈ is closed under arbitrary unions,
iii.
 Ʈ is closed under finite intersections.
 
The family Ʈ is said to be a topology on set
X. Members of Ʈ are said to be open in X or
open subsets of X.
 
Indiscrete topology
: The topology Ʈ on the set X
consist of only ɸ and X.
 The 
Indiscrete topology
is induced by the Indiscrete pseudo- metric on X.
Discrete topology 
: Here the topology coincides
with the power set P(X). The discrete topology is
induced by the discrete metric.
Co-finite topology 
: A subset A of X is said to be
co-finite, if its complement, X-A is finite. Let Ʈ
consists of all co-finite subsets of X and the
empty set. In the case X is finite it coincides with
the discrete topology but otherwise it is not the
same.
 
Co-countable topology 
: The co-countable
topology on a set is defined by taking the
family of all sets whose complements are
countable and the empty set.
The usual topology 
: The usual topology on R
is defined as the topology induced by the
Euclidean metric.
 
DEFINITION
The topology Ʈ
1
 is said to be 
weaker
 (or
coarser
) than the topology Ʈ
2
  (on the same
set) if Ʈ
1
 
Ϲ
 
Ʈ
2
 as the subsets of the power set.
THEOREM
 
Let X be a set {Ʈ
1
:i 
ϵ 
I
} be an indexed family of
topologies on X. let Ʈ=
 
 Then Ʈ is a topology on X. It is weaker than
each Ʈ
i 
,i 
ϵ 
I
. If 
Ư
 
is a any topology on X which
is weaker than each Ʈ
i 
,i 
ϵ 
I
,then Ʈ is stronger
than 
Ư
 .
 
 
 
Let X be a  set and  
a
 
family of subsets of X.
Then there exists a unique topology Ʈ on X,
such that it is the smallest topology on X
containing 
.
 
BASES AND SUB -BASES
 
DEFINITION
 
Let (X,
 Ʈ
) be a topological space. A subfamily Ḅ of 
Ʈ
is said to be a base for 
Ʈ
 if every member of 
Ʈ
can be expressed as the union of some members of Ḅ.
PREPOSITION
 
Let (X,
 Ʈ
 ) be a topological space and Ḅ 
Ϲ
 
Ʈ
 . Then Ḅ is a
base for 
Ʈ
 iff for any x
ϵ
X and any open set G
containing x, there exists B 
ϵḄ
 such that x
 ϵB
and B containing G.
 
 
A space is said to satisfy the second axiom of
countability or is said to be second countable
if its topology has  a countable base.
THEOREM
 
If a space is second countable then every
open cover of it has a countable subcover.
 
 
 PROPOSITION 1:
 
Let 
Ʈ
1
 
,
 
Ʈ
2
  be two topologies for a set having bases Ḅ
1
2
respectively. Then 
Ʈ
1
 
is weaker than 
 
Ʈ
2
 
iff every
member of Ḅ
1
 can be expressed as a union of some
members of Ḅ
2
.
  
 
PROPOSITION 2:
 
Let X be a set and Ḅ a family of its subsets covering X.
Then the following statements are equivalent :
(1) There exists a topology on X with Ḅ as base.
(2) for any Ḅ
1 ,
2
 ϵ
 Ḅ and x
 ϵ
1 
n
2
 
there exists Ḅ
3
 
ϵ
such that x
 ϵ
3
 and Ḅ
3
 contain Ḅ
1 
n
2
 
  (3) for any
1 ,
2
 ϵ
 Ḅ ,
 
1 
n
2
 
can be expressed as
the union of some members of 
 
 
Let X be a set, Ʈ a topology on X and  Ș a
family of subsets of X. Then  Ș  is a sub-base
for Ʈ iff Ș  generates Ʈ
Given any family Ș of subset of X , there is a
unique topology Ʈ on X having Ș as a sub-
base. Further, every member of Ʈ can be
expressed as the union of sets each of which
can be expressed as the intersection of
finitely many members of Ș .
 
K D JOSHI- INTRODUCTION TO GENERAL
TOPOLOGY (SECOND EDITION) ,NEW AGE
INTERNATIONAL PUBLISHERS
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Topology is the study of surfaces derived from Greek words meaning surface and discourse. In mathematics, a topological space is a set with a family of subsets satisfying specific properties. Examples include the open ball, open set, indiscrete topology, discrete topology, co-finite topology, and co-countable topology. Different topologies can be compared in terms of strength. Understanding these fundamental concepts is crucial for advanced studies in mathematics.


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  1. Prof.Retheesh R Department of Mathematics

  2. INTRODUCTION TOPOLOGICAL SPACES EXAMPLES OF TOPOLOGICAL SPACES BASES AND SUB BASES REFERENCES

  3. The word Topology Greek words topos meaning surface and logos meaning discourse or study . Topology thus literally means study of surfaces. Topology is derived from the two

  4. Definitions Open ball: Let x0 X and r be a positive real number. Then the open ball with centre x0 and radius r is defined to be the set { x X: d(x, x0)<r } which is denoted either by Br(x0) or by B(x0,r). It is also called open r ball around x0. Open set: A subset A X is said to be open if for every x0 A there exists some open ball around x0 which is contained in A, that is ,there exists r>0 such that B(x0,r) A.

  5. TOPOLOGICAL SPACE A topological space is a pair (X , ) where X is a set and is a family of subsets of X satisfying. i. and X ii. is closed under arbitrary unions, iii. is closed under finite intersections. X. Members of are said to be open in X or open subsets of X. The family is said to be a topology on set

  6. Indiscrete topology: The topology on the set X consist of only and X. The Indiscrete topology is induced by the Indiscrete pseudo- metric on X. Discrete topology : Here the topology coincides with the power set P(X). The discrete topology is induced by the discrete metric. Co-finite topology : A subset A of X is said to be co-finite, if its complement, X-A is finite. Let consists of all co-finite subsets of X and the empty set. In the case X is finite it coincides with the discrete topology but otherwise it is not the same.

  7. Co-countable topology : The co-countable topology on a set is defined by taking the family of all sets whose complements are countable and the empty set. The usual topology : The usual topology on R is defined as the topology induced by the Euclidean metric.

  8. DEFINITION The topology 1 is said to be weaker coarser set) if 1 2 as the subsets of the power set. THEOREM Let X be a set { 1:i I} be an indexed family of topologies on X. let = Then is a topology on X. It is weaker than each i ,i I. If is a any topology on X which is weaker than each i ,i I,then is stronger than . DEFINITION weaker (or coarser) than the topology 2 (on the same THEOREM

  9. Let X be a set and a family of subsets of X. Then there exists a unique topology on X, such that it is the smallest topology on X containing .

  10. BASES AND SUB -BASES DEFINITION Let (X, ) be a topological space. A subfamily of is said to be a base for if every member of can be expressed as the union of some members of . PREPOSITION Let (X, ) be a topological space and . Then is a base for iff for any x X and any open set G containing x, there exists B such that x B and B containing G.

  11. A space is said to satisfy the second axiom of countability or is said to be second countable if its topology has a countable base. THEOREM If a space is second countable then every open cover of it has a countable subcover.

  12. PROPOSITION 1: Let 1, 2 be two topologies for a set having bases 1 2 respectively. Then 1 is weaker than 2 iff every member of 1 can be expressed as a union of some members of 2. PROPOSITION 2: Let X be a set and a family of its subsets covering X. Then the following statements are equivalent : (1) There exists a topology on X with as base. (2) for any 1 , 2 and x 1 n n 2 there exists 3 such that x 3 and 3 contain 1 n n 2 (3) for any 1 , 2 , 1 n n 2 can be expressed as the union of some members of

  13. Let X be a set, a topology on X and a family of subsets of X. Then is a sub-base for iff generates Given any family of subset of X , there is a unique topology on X having as a sub- base. Further, every member of can be expressed as the union of sets each of which can be expressed as the intersection of finitely many members of .

  14. K D JOSHI- INTRODUCTION TO GENERAL TOPOLOGY (SECOND EDITION) ,NEW AGE INTERNATIONAL PUBLISHERS

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