Definition of Topology
Let X be a non empty set. A collection of subsets of X is said to be a topology on X if the following
conditions are satisfied:
(i) The union of any number of members of belongs to .
(ii) The intersection of finite number of members of belongs to .
(iii) The empty set
and set X itself belongs to .
In other words, a collection
of subsets of a nonempty set X is said to be topology on X if it is
closed under the formation of arbitrary unions and finite intersections and contains both
and X.
The sentence “closed under the formation of arbitrary unions” means that the union of any number
of members of is in . Similarly, the sentence “closed under the formation of finite intersections” means
that the intersection of any finite number of members of is in .
If is a topology on X, then the pair (X, ) is called atopological space. The set X is called the
underlying set or the ground set and the elements of the set X are called the points of the topological space.
Instead of writing (X, ), we may write X for a topological space if there is no danger of confusion. In fact,
Mathematicians do not bother about specifying the topology. If it is said that X is a topological space then
one should himself understand that there is a topology defined on the set X.
Example:
Let X = {a, b, c} and consider the collections
={ , {a}, {c}, {ac}, X },
={ , {a}, {a,b}, {a,c}, X } are the topologies on X.
Indiscrete and Discrete Topology
Indiscrete Topology:
The collection of the non empty set and the set X itself is always a topology on X and is said to
be indiscrete topology on X. In other words, for any non empty set X, the collection
indiscrete topology on X, and the space
indiscrete space.
is an
is called the indiscrete topological space or simply an
Discrete Topology:
The power set P(X) of a non empty set X is said to be discrete topology on X and the
space(X,P(X)) is called the discrete topological space or simply a discrete space.
Now we shall show that the power set of a non empty set X is a topology on X. For this
let
be the power set of X, i.e. the collection of all possible subsets of X, then
(i) The union of any number of subsets of X, being the subset of X, belongs to .
(ii) The intersection of finite number of subsets of X, being the subset of X, belongs .
(iii)
and X, being the subsets of X, belongs to .
This shows that the power set is a topology on X.
Coarser and Finer Topology
If
of
and
is also in
stronger than
are two topologies defined on the non empty set X such that
, then
is said to be coarser or weaker than
and
, i.e. each member
is said to be finer or
.
It may be noted that indiscrete topology defined on the non empty set X is the weakest or coarser
topology on that set X and discrete topology defined on the non empty set X is the stronger or finer topology
on that set X.
Note: The topology which is both discrete and indiscrete such topology which has one element in set X.
i.e. X = {a},
{ , X}. Every singleton set is discrete as well as indiscrete topology on that set.
Intersection of Topologies
Intersection of any two topologies on a non empty set is always topology on that set. While the
union of two topologies may not be a topology on that set.
Example:
Let
is a topology on X.
is not a topology on X.
Given two (and in fact any number of) topologies
weaker than both
both
and
and
, is contained in both
is a topology which contains both
We write
and
and
on X there is a topology
which is
contains every topology on Xwhich is weaker than
.
Similarly there is a topology
union
,
which contains both
and
and call
then
and
is the weakest in the sense that if
.
as the topology generated by
which may not be a topology. Here
and
.
is different from the
is the set theoretic union of the collection
.
Open Subset of a Topological Space
Let
topological space
be a topological space, then a member of
, the members of
always the member of , so
is said to be an open set in X. Thus, in a
are said to be open subsets of X. Since
and X are always open sets in X.
and full space X are
On the other hand we can define as let
said to be an open set of X (or in X), if
.
Example: If
sets of X.
with topology
is the topological space, then the subset
, then
and
of X is
are the possibleopen
On the other hand if
, then
is not an open set of X. It is clear from this
illustration that the open subsets of a space X depend upon the topology defined on X.
Theorems:





Every subset of a discrete topological space is open.
The union of any numbers of open subsets of a topological space is open.
The intersection of any finite number of open subsets of a topological space is open.
If Y is an open subspace of a topological space X, then each open subset of Y is also open inX.
Every subset of a topological space is open if and only if its each singleton subset is open.
Closed Subset of a Topological Space
Let
be a topological space, then a subset of Xwhose complement is a member of
be a closed setin X. Thus, in a topological space
be closed subsets of X. Since
If
subsets of X.
(
are said to
and the full space X are always closed sets of X.
On the other hand we can define as let
to be closedin X, if
, the complements of the members of
is said to
be a topological space, then the subset
of X is said
is open in X).
with topology
, then
and
are the possible closed
Remark:
If
with topology
neither closed nor open set in X. The subset
, the subset
of X is such that
is also neither open nor closed in X.
In general, the discrete topological space
there does not exist any subset of X, which is
neither open nor closed in X, i.e. all the subsets in discrete topological space are open as well as closed.
Clopen Set:
Let
be a topological space. A subset of X which is open as well as closed is said to be clopen
set. Since and X are open as well as closed, so there are clopen sets. Since each subset of a discrete
topological space is open as well as closed so each subset of a discrete topological space is a clopen set.
Theorem:
o
o
o
o
Every subset of a discrete topological space is closed.
The intersection of any number of closed subsets of a topological space is closed.
The union of any finite number of closed subsets of a topological space is closed.
Every subset of a discrete topological space is clopen.
Usual Topology on Real
Usual Topology on :
A collection of subsets of
which can be can be expressed as union of open intervals, forms a
topology on , and is called topology on .
Remark:
Every open interval is an open set but the converse may not be true.
Usual Topology on
:
Consider the Cartesian plane
, then the collection of subsets of
which can be expressed as
union of open discs or open rectangles with edges parallel to coordinate axis from a topology, and is called
usual topology on
.
Usual Topology on
:
Consider the Cartesian plane
, then the collection of subsets of
which can be expressed as
union of open spheres or open cubed with edges parallel to coordinate axis from a topology, and is called
usual topology on
.
Cofinite Topology
Let X is a non empty set, and then the collection of subsets of X whose compliments are finite along
with
(empty set), forms a topology on X, and is called co-finite topology.
Example:
Let
with topology
topology because the compliments of all the subsets of X are finite.
is a co-finite
Note:
It may be noted that every infinite set may or may not be co-finite topology, for this
suppose
(set of real numbers which is infinite set) with
topology
is a co-finite topology because compliments of all the
members of topology along with empty set are finite.
Remark:
If X is finite, then topology is discrete. For a subset of X belongs to if and only if, it is either
empty or its compliment is finite. When X is finite, the compliment of each of its subset is finite and
therefore, each subset of X belongs to . Hence
is the discrete topology on X.
Subspaces of Topology
We shall describe a method of constructing new topologies from the given ones. If
is a
topological space and
is any subset, there is a natural way in which Y can “inherit” a topology from
the parent set X. It is easy to verify that the set
, as runs through , is a topology on Y. This
prompts the following definition of (relative topology).
Relative Topology or Inherited Topology:
Let
be a topological space and
a topology on Y, called the topology induced by
called the subspace of X. The topology
In other words, if
a nonempty subset, then
is
on Y or relative topology on Y. The pair
is
is also called the inherited topology.
is a topological space and Yis a non empty subset of X. The collection
consisting of those subsets of Y which are obtained by the intersections of the members of
called the relative topology onY. It is clear from the definition of the relative topology,
with Y is
, that each of its
member is obtained by the intersection of some members of
with Y. It should be noted that not every
subset Y of X is a subspace of X. The subset Y of X is a subspace of X if and only if the topology of Y is the
relative topology.
Example:
Let
and
topology on Y will be
with topology
, using the definition of relative topology
generated the
is a relative topology.
Remark:
Let
be a topological space and Y is the subset of X. Then every open subsets of Y is also open
in X, if and only if, Y itself is open in X. In other words the subspace of a discrete topological space is also
discrete space.
Limit Point of a Set
Let X is a topological space with topology , and A is a subset of X. A point
is said to be
the limit point oraccumulation point or cluster point of A if each open set containing contains at least
one point of A different from .
In other words, a point of a topological space X is said to be the limit point of a subset A of X if
every open set containing , we have
It is clear from the above definition that the limit point of a set A may or may not be the point of A.
Let
with topology
and
point of A, because the open sets containing b namely
, then
is the only limit
and X also contains a point a of A.
On the other hand, a and b are not limit point of
, because the open set
containing
these points do not contain any point of C. The point c is also not a limit point of C, because the open
set
containing c does not contain any other point of C different from c. Thus, the set
limit points.
As another example, let
with
topology
. Let
because the open set
has no
then a is not a limit point of A,
containing a does not contain any other point of A different from a. b is a limit
point of A, because the open sets
and X containing b also contain a point of Adifferent from b.
Similarly, d and e are also limit points of A. This illustration suggests that a set can have more than one
limit point.
Derived Set:
Let
be a topological space, and let A be a subset of X. The set of all limit points of Ais said to
be the derived set and is denoted by
Remark:
or
. In the above example,
.
It may be noted that under usual topology, consider the subsets
real, then all the points of these intervals are limits points.
,
,
,
of
Isolated Point of a Set
Let A be a subset of a topological space X, then a point
is said to be an isolated point of A if
there exist an open set containing which does not contain any point of Adifferent from . In other words,
a point
is said to be an isolated point of A if there exist an open set U containing such
that
. It is obvious from the definition of an isolated point of a set that an isolated point
of A can never be the limit point of A. The set of all isolated points of A is usually denoted by
.
Theorem: Any closed subset of a topological space X is the disjoint union of its set of isolated points and
its set of limit points in the sense that it contains these sets, they are disjoint, and it is their union.
Perfect Set:
A subset of a topological space is said to be perfect set if it is equal to its derived set. Thus, a
subset A of a topological space X is said to be a perfect set if
.
Theorem: A subset of a topological space is perfect if and only if it is closed and has no isolated points.
Proof: Let A be a perfect subset of a topological space X, then
. Since
is the set of all limit
points of A and a limit point is not an isolated point, so A has no isolated points.
Conversely, Let A be closed, and has no isolated points, then A is equal to its derived set,
i.e.,
, so A is perfect.
Closure of a Set
Let
be a topological space and A is a subset ofX, then the closure of A is denoted by
or
is the intersection of all closed sets containing A or all closed super set of A. i.e. the smallest
closed set containing A.
On the other hand it can also be as let
be a topological space and let A be any subset of X. A
point
is said to be adherent to A if each neighborhood of contains a point of A (which may be
itself). The set of all points of X adherent to A is called closure (or adherence) ofA and is denoted by
symbols
. In
Remarks:


Every set is always contained in its closure. i.e.
Closure of a set by definition (being intersection of closed set is always closed set).
Example:
Let
with topology
and
be a subset
of X.
Open sets are
Closed sets are
Closed sets containing A are
Now
Theorem: Let

be a topological space, and A and B be subsets of X, then
A is closed if and only if




Dense Subset of a Topological Space:
Let
dense in X), if
be a topological space and A be a subset of X, then A is said to be dense subset of X. (i.e.
Example: Consider the set of rational number
containing
in
. Which shows that
(with usual topology), then the only closed set
. Hence
is dense in
.
Remarks:
o
o
It may be noted that the set of irrational numbers is also dense in . i.e.
Rational are dense in and countable but irrational numbers are also dense in
countable.
.
but not
Neighbourhood of a Point
Let
be a topological space. A subset N of X containing
is said to
beneighbourhood of if there exist an open set U containing such that N contains U, i.e.
A neigbourhood of a point is not necessarily an open set. However, if a neighbourhood of a point is an open
set, we call it an open neighbourhood of that point.
If
with topology
(known as Sierpinski space), then
neighbourhood of , because, we can find an open set
and Xare
such that
and
On the other hand, X is the only neighbourhood of , because, we can find the open set Xsuch that
As another example, let
then
Similarly,
the only neighbourhood of
with topology
are neighbourhood of .
are neighbourhoods of , and X is
and . It is clear from this illustration that a point
may have more than one
neighbourhood.
Neighbourhood System:
Let
be a topological space. The set of all neighbourhoods of a point
a neighbourhood system of . It is denoted by
system.
is said to be
. The above example shows that neighbourhood
Theorems:
o
o
o
o
o
o
o
o
o
o
The topological space X itself is a neighbourhood of each of its points.
A subset of a topological space is open if and only if it is the neighbourhood of each of its
own points.
The intersection of any two neighbourhoods of a point is also its neighbourhood in a
topological space.
The union of any two neighbourhoods of a point is also its neighbourhood in a topological
space.
If A is a neighbourhood of point x, and
, then show that B is also neighbourhood of
point x.
If A is a neighbourhood of point x, then show that there exists open set B such that B is also a
neighbourhood of point x and A is a neighbourhood of each point of B.
The neighbourhood system of a point is a non empty set.
The intersection of a finite number of neighbourhoods of a point is also its neighbourhood.
Any subset M of a topological space X which contains a member of N(x) also belongs
to N(x).
Each neighbourhood of a point of a cofinite topological space is open.
Interior Point of a Set
Let
be the topological space and
point of set A, if there exist an open set such that
, then a point
In other words let A be a subset of a topological space X, a point
points of A if is in some open set contained in A.
Interior of a Set:
is said to be an interior
is said to be an interior
Let
be a topological space and A be a subset ofX, then the interior of A is denoted
by
or
is defined to be the union of all open sets contained in A.
In other words let
be a topological space and A be a subset of X. The interior of A is union of
all open subsets of A, and a point in the interior of A is called an interior point of A.
Remarks:

Interior of A is union of all open sets contained in A. Union of open sets is again an open set. Hence
interior of A is the largest open set contained in A.


and
Interior of sets is always open.

Example:
Let
If
with topology
.
, then find
.
Since there is no open set containing
Similarly,
and a subset of A, so
is not an interior point of A. Since
is not an interior point of A.
is n open set containing
and is a subset of A, so
interior point of A. Hence
Theorems:
o
o
o
Each point of a non empty subset of a discrete topological space is its interior point.
The interior of a subset of a discrete topological space is the set itself.
The interior of a subset A of a topological space X is the union of all open subset of A.
o
The subset A of topological space X is open if and only if
o
If A is a subset of a topological space X, then
o
Let
then (1)
.
.
be a topological space and A and B are subsets of X,
(2)
(3)
is an
Exterior Point of a Set
Let
be a topological space and A be a subset of X, then a point
an exterior point of A, if there exist an open set U, such that
, is said to be
In other words, let A be a subset of a topological space X. A point
point of A if there exists an open set U containing
such that
is said to be anexterior
.
Exterior of a Set:
The set of all exterior points of A is said to be the exterior of A and is denoted by
.
Remark:
It may be noted that an exterior point of A is an interior point of
.
Theorems:

If A is a subset of a topological space X, then (1)

If A is a subset of a topological space X, then

In a topological space X, (1)

If A is a subset of a topological space X, then (1)

If A is a subset of a topological space X, then
(2)
.
.
(2)
.
(2)
is the largest open subset of
.
.
Boundary Point of a Set
Let A be a subset of a topological space X, a point
point of A if each open set containing at
intersects both
is said to be boundary point orfrontier
and
.
The set of all boundary points of a set A is called the boundary of A or the frontier of A. It is
denoted by
. Since, by definition, each boundary point of A is also a boundary point of
versa, so the boundary of A is same as that of
, i.e.
and vice
.
Theorems:

If A is a subset of a topological space X, then
.

If A is a subset of a topological space X, then
.





If A is a subset of a topological space X, the A is open
.
A subset of a topological space X is closed if and only if it contains its boundary.
A subset of a topological space has empty boundary if and only if it is both open and closed.
The boundary of a closed set is nowhere dense in a topological space.
Let X be a topological space. Then any closed subset of X is the disjoint union of its interior and its
boundary, in the sense that it contains these sets, they are disjoint, and it is their union.