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Base or Open Base of a Topology
Let
be a topological space, then the sub collection of is said to be base or basesor open
base for , if each member of can be expressed as union of members of .
In other words let
for a point
be a topological space, then the sub collection
belonging to an open set U then there exist
of
such that
is said to be base, if
.
Example:
Let
onX.
each member of
and let
be a topology defined
is a sub collection of
is a union of members of .
, which meets the requirement for a base, because
Remarks:

It may be noted that there may be more than one base for a given topology defined on that set.

Since union of empty sub collection of members of
is an empty set, so empty set
.
For Discrete Topology:
Let
and let
defined on X.
of
there must becomes .
be a topology
is a base for . Check weather
is a base or not take all possible union
Possible Unions
In this case discrete topological space, the collection of all singletons subsets of X forms a base for discrete
topological space.
For Indiscrete Topology:
Let
and let
be a topology defined on X.
is a base for .
Theorem:
Let
be a topological space, then a sub collection
of
is a base for
if and only if
o
o
If
and
belongs to
for
, then
then there exist
can be written as union of members of
of
such that
. i.e.
.
Subbase for a Topology
Let
be a topological space. A sub-collection of subset of is said to be an open subbase
for or a subbase for topology if all finite intersection of members of forms a base for .
In others words, A class of open sets of a space
is called a subbase for a topology on , if
and only if intersections of members of forms a base for topology on . The topology obtained in this
way is called the topology generated by .
Example:
Consider the Cartesian plane
with usual topology the
be the base for the topological space
, then the collection of all intervals of the form
,
where
and
gives a
subbase for . Since the finite intersection of all such intervals gives the members of the base of ,
i.e.,
.
Example:
Let
with topology
is a subbase for
.
Theorems:


Let be a non-empty collection of subsets of
. Suppose that
, then is a subbase for
some topology on
.
Let
be any non-empty set, and let be an arbitrary collection of subsets of . Then can
serve as an open subbase for a topology on
, in the sense that the class of all unions of finite
intersections of sets in is a topology.
Local Base for a Topology
Let
at a point
be a topological space and
, if
belonging to an open set
It can be defined as, let
to be neighbourhood base at a point
open set
containing
, there is a
, then the sub collection
, there exist a member
of
is said to be local bases
, such that
.
be a topological space and
. A sub collection
of is said
or local bases at a point or simply a base at a point , if for any
such that
.
Remark:


It may be noted that every bases for a topology is also a local base at each point of ground set but
the converse may not be true.
Union of all local bases forms bases for topology defined on the any non-empty set X.
Example:
Let
be a non-empty set with topology
on X. Consider the open sets containing “a” are
at point “a”.
Since
“a”. Similarly,
Now
then
,
Note that
and
is a local base
.
. Which shows that
and
defined
is also a local bases at point
.
which forms a bases for
.
It may noted that above procedure of finding local bases is only valid when a number of open sets
containing a point is finite.
Example:


Consider
(Cartesian plane) with usual topology, and let x be any point of
, then collection of
all open discs with centre at x form a local bases at x.
Every discrete topological space has a countable neighbourhood base at each of its points.
First Countable Space
Let
be a topological space, then
a countable local bases, i.e., if every
,
is said to be first countable space if for every
has
is countable.
In other words, a topological space
is said to be the first countable space if every point
of has a countable neighbourhood base. A first countable space is also said to be a space satisfying the
first axiom of countability.
Example:
If
is finite, then
is local base of
, then
is first countable space. As
is also finite. So,
is finite, so its every subset is finite. If
is first countable space.
Example:
Let
and
be a non-empty set
be topology defined on
Base at “a”
Base at “b”
Base at “c”
Base at “d”
.
Here each local base is countable, so
is first countable space.
Example:
If
is either countable or uncountable and
first countable space, because for each
finite.
,
and
,
is discrete topology on
, then
always
(singleton) is the local base and so local base is
is countable (finite).
Second Countable Space
Let
countable bases.
be a topological space, then
is said to be second countable space, if
has a
In other words, a topological space
is said to be second countable space if it has a countable
open base. A second countable space is also said to be a space satisfying the second axiom of countability.
Example:
If
is finite, then member of each
countable space. Now we show that
(Countable), then
Therefore,
countable space.
on
is finite. So its base is finite. Hence
is first countable. Let
(countable), so
is subbase of
is second
. So,
is also countable.
is second countable space, as each local base is also countable, so this is also first
Theorems:




Every second countable space is first countable space, but the converse may not be true.
Any uncountable set
with co-finite topology is not first countable and so is not second countable.
The set of all intervals with rational ends is a countable base for the usual topology on . The real
line is second countable space.
Any uncountable set
with countable topology is not first countable and so is not second
countable.
Lindelof Space
Open Cover:
Let
be a topological space. A collection
of open subsets of
is said to be
open cover for
if
.
A sub-collection of an open cover which is itself an open cover is called a sub-cover.
Lindelof Space:
A topological space
sub-cover.
is said to be a Lindelof space if every open cover of
has a countable
Theorems:


A closed sub-space of Lindelof space is Lindelof.
Every second countable space is Lindelof space.
Lindelof Theorem:
Let
be a second countable space. If a non-empty open set
a collection
of open sets, then
in
is represented as the union of
can be represented as a countable union of
.
Separable Space
i.e.,
(1)
A topological space
, is said to be separable space, if it has a countable dense subset in
,
, or
In other words, a space
, where
is an open set.
is said to be separable space if there is a subset
is countable (2)
(
is dense in
).
of
such that
.
Example:
Let
be a non-empty set and
topology defined on
are
in
is a
. Suppose a subset
. The closed set
. Now we have
. So,
. Since
is finite and dense
is a separable space.
Example:
Consider the set of rational number
a subset of
(with usual topology), then the only closed set
containing
is
which shows that
. Since
is dense in
set of irrational numbers is dense in
but not countable.
, so
is also separable in
. But
Theorems:
o
o
o
o
Every second countable space is a separable space.
Every separable space is not second countable space.
Every separable metric space is second countable.
The continuous image of a separable space is separable.
Continuity in Topological Spaces
Let
be a function define from topological space
continuous at a point
such that
if every neighbourhood
set
, such that
, then
is said to be
, there exist a neighbourhood
of
,
.
In other words, let
at a point
of
to topological space
be two topological spaces. A function
if and only if for every open set
;
is said to be continuous
, which contains
is the inverse image of
, there exist an open
.
It can also be defined as, let
and
is said to be a continuous function at a point
there is a neighbourhood
a continuous function on
be topological spaces. A function
of
if for any neighbourhood
of
in
such that
if it is continuous at each point of
of
in
. The function
,
is said to be
.
Note: It may be noted that a function from topological space
continuous on
if
is continuous of each points of
.
to topological space
is said to be
Theorems:
o
A function
from one topological space
if and only if for every open set
o
A function
in
o
If
and
,
is open in
from one topological space
if and only if for every closed set
in
,
of
,
If
o
only if for any sub set
of ,
If
is an arbitrary topological space and
are topological spaces, then a function
function
Let
be topological spaces. If
mappings, then
Open Mapping and Closed Mapping
Open Mapping:
.
is continuous on
if and
is continuous on
if and
.
is an indiscrete topological space, then every
is a continuous function on
and
is continuous
.
o
o
is closed in
is continuous
.
into another topological space
are topological spaces, then a function
only if for any sub set
and
into another topological space
is continuous.
.
and
are continuous
A mapping
from one topological space
into another topological space
is said to be an open
mapping if for every open set
in ,
is open in . In other words a mapping is open if it carries
open sets over to open sets. A continuous mapping may not be open.
Example: Let
with topology
topology
, then
but not open, because
and let
defined by
is open in
,
but
with
,
is not open in
is continuous
.
Closed Mapping:
A mapping
from one topological space
closed mapping if for every closed set
in
if it carries closed sets over to closed sets.
into another topological space
,
is closed in
is said to be an
. In other words a mapping is closed
Theorems:

Let
and
if

Let
Let
is open if and only
is continuous.
and
if

be topological spaces. A bijective mapping
be topological spaces. A bijective mapping
is closed if and only
is continuous.
and
if
be topological spaces. A mapping
for every open subset
of
is open if and only
.
Homeomorphism
A function
is said to be homeomorphism (topological mapping) if and only if the
following conditions are satisfied:
(1)
is bijective.
(2)
(3)
is continuous.
is continuous.
It may be noted that if
is a homeomorphism from
to , then
is said to be homeomorphic to
and is denoted by
. Form the definition of a homeomorphism, it follows that
and
are
homeomorphic spaces, then their points and open sets are put into one-to-one correspondence. In other
words,
and
differ only in the nature of their points, but from the point of view of the subject of
topology, they are identical or have the same topological structure.
Remarks: : “Homeomorphism” helps reducing the complicated problems in simple form, that is, an
apparently complicated space may possibly be homeomorphic to some space more familiar to us. Hence in
this way, one determines the properties of complicated spaces easily.
Theorems:
o
o
Bijective continuous mapping
is open if and only if
If
and
are topological spaces, let
means that
and
Then this relation is reflexive, symmetric and transitive.
o
Let
and
be topological spaces and
following are equivalent. (1)
is open in
in
if and only if
if and only if
is continuous.
are homeomorphic.
be a bijective function, then the
is a homeomorphism. (2) For any subset
is open in
is closed in
. (3) For any subset
. (4) For any subset
of
of
,
of
,
,
is closed
.
Topological Property
A property P is said to be a topological property if whenever a space
has the property P, all
spaces which are homeomorphic to
also have the property P,
.
In other words, a topological property is a property which if possessed by a topological space is also
possessed by all topological spaces homeomorphic to that space.
Note: It may noted that length, angle, boundedness, Cauchy sequence, straightness and being triangular or
circular are not topological properties. Whereas limit point, interior, neighbourhood, boundary, first and
second countability, separablility are toological properties. We shall come across several topological
properties in the sequel. Because of its critical role the subject topology is usually described as the study of
topological properties.
Examples:
o
Let
and
defined by
. Then
is a
homeomorphism and therefore
. Note that
and
have different
lengths, therefore “length” is not a topological property. Also
is bounded and
is not
bounded, therefore “boundeness” is not a topological property.
o
Let
defined by
sequences
o
o
, then
and
is homeomorphism. Consider the
in
.
sequence, where
is not. Therefore, “being a Cauchy sequence” is not a topological
property.
Straightness is not topological property, for a line may be bent and stretched until it is
wiggly.
Being “triangular” is not a topological property since a triangle can be continuously
deformed into a circle and conversely.
Product Topology
Products of Sets:
If
is a Cauchy
and
are two non-empty sets, then Cartesian product
is defined
as
.
Projection Maps:
Let A and B be non-empty sets, then it can be defined the following two functions
(1)
defined as
for all
(2)
defined as
for all
The above maps are called the projection maps on A and B respectively.
Note: Let
similarly.
be non-empty sets, then the projection maps
can be defined
Product Topology:
Let
on
be the product of topological spaces
and
. The coarsest topology
with respect to which the projection maps
and
continuous, is said to be product topology and thus the space
are
is said to be the product space.
Remarks:

It may be observed that if
and
are distinct topological spaces then the
collection

form a subbase for product topology
on
.
It may be noted that if A and B are any open interval, then
will be open rectangle strips.
Collection of open rectangle form a bases for usual topology on
product of finite number of spaces
then
and
. So, generalizing this fact to the
are topological spaces
form a bases for product topology.