Download T0 Space A topological space X is said to be a T0

T0 Space
A topological space X is said to be a T0-space if for any pair of distinct points of X, there exist at
least one open set which contains one of them but not the other.
In other words, a topological space X is said to be a T0-space if for any
exist an open set
such that
but
, there
.
Example:
Let
space, because
with topology
defined on X, then

for a and b, there exist an open set
such that
and

for a and c, there exist an open set
such that
and

for b and c, there exist an open set
such that
and
is a T0-
Theorems:




Let X is an indiscrete topological space with at least two points, then X is not a T0-space.
Let X is a discrete topological space with at least two points, then X is not a T0-space.
The real line
with usual topology is T0-space.
Every sub space of T0-space is T0-space.
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
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A topological space X is T0-space if and only if, for any
Every two point co-finite topological space is a T0-space.
Every two point co-countable topological space is a T0-space.
If each singleton subset of a two point topological space is closed, then it is T0-space.
If each finite subset of a two point topological space is closed, then it is a T0-space.
Any homeomorphic image of a T0-space is a T0-space.
A pseudo metric space is a metric space if and only if it is a T0-space.
.
T1 Space
A topological space X is said to be a T1-space if for any pair of distinct points of X, there exist two
open sets which contain one but not the other.
In other words, a topological space X is said to be a T1-space if for any
there exist
open sets
and
such that
and
.
Example:
Let
with topology
because for
, we have open sets
find an open set which contains c but not a, so
that
true.
defined on X is not a T1-space
and X such that
. This shows that we cannot
is not a T1-space. But we have already showed
is a T0-space. This shows that a T0-space may not be a T1-space. But the converse is always
Example:
The real line
with usual topology is T1-space.
To prove this, we suppose that
topology on
that
. Further assume that
consists of open intervals, so we have open sets
and
. This shows that the real line
. Since the usual
and
, such
with usual topology is a T1-space.
Theorems:
o
o
o
o
o
o
o
o
o
o
o
Every T1-space is a T0-space
An indiscrete topological space with at least two points is not a T1-space.
The discrete topological space with at least two points is a T1-space.
Every two point co-finite topological space is a T1-space.
Every two point co-countable topological space is a T1-space.
Every subspace of T1-space is T1-space.
A topological space is a T1-space if and only if its each finite subsets is a closed set.
Following statements about a topological space X are equivalent. (1) X ia a T1-space. (2)
Each singleton subset of X is closed. (3) Each subset A of X is the intersection of its open
supersets.
Any homeomorphic image of a T1-space is a T1-space.
If x is a limit point of a set A in a T1-space X, then every open set containing an infinite
number of distinct points of A.
A finite set has no limit points in a T1-space.
T2 Space or Hausdorff Space
A Hausdorff space or T2-space is a topological space in which each pair of distinct points can be
separated by disjoint open set.
In other words, a topological space x is said to be a T2-space or Hausdorff space if for
any
, there exist open sets
and
such that
and
.
Example:
Let
be a non-empty set with topology
discrete topology). Hence
(all the subsets of X, powers set or
For
For
For
and
is a T2-space.
For
For
For
Theorems:
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Every metric space is Hausdorff space.
Every T2-space is a T1-space but the converse may not be true.
Every subspace of T2-space is T2-space.
In a Hausdorff space, any point and a disjoint compact subspace can be separated by open sets, in
the sense that there exist disjoint open sets one contains the point and the other contains the compact
subspace.
Every compact subspace of Hausdorff space is closed.
A one-to-one continuous mapping of a compact onto a Hausdorff space is homeomorphism.

Let

suppose that
for all x in a dense subset D of X. Then
for all x in X.
A space X is Hausdorff space if and only if every point a of X is the intersection of its closed
neighbourhoods.

Let X be a topological space and Y a Hausdorff space. Let
,
from X to Y, then the set
be continuous functions from a space X to a Hausdorff space Y and
and
be continuous function
is a closed set.
Regular Space
Let
be a topological space, then for every non-empty closed set
and a point x which does
not belongs to , there exist open sets
and , such that
and
In other words, a topological space X is said to be a regular space if for any
set A of X, there exist an open sets
and
such that
and
.
and any closed
.
Example:
Show that a regular space need not be a Hausdorff space.
For this, let X be an indiscrete topological space, then the only non-empty closed set X, so for
any
, there does not exist and closed set A which does not contain x. so X is trivially a regular space.
Since for any
Hausdorff space.
, there is only one open set X itself containing these points, so X is not a
T3-Space:
A regular T1-space is called a T3-space.
Theorems:
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Every subspace of a regular space is a regular space.
Every T3-space is a Hausdorff space.
Let X be a topological space, then the following statements are equivalent. (1) X is a regular space.
(2) For every open set U in X and a point
there exist an open set Vsuch
that
sets.
. (3) Every point of X has a local neighbourhood basis consisting of closed
Completely Regular Space
A topological space X is said to be completely regular space, if every closed set A in X and a
point
and
,
, then there exist a continuous function
, such that
.
In other words, a topological space X is said to be a completely regular space if for any
closed set Cnot containing x, there exist a continuous function
such that
and a
and
.
Remark:
Let us consider a continuous function
defined as
constant function is continuous therefore taking the function
where
and
. Since
,
.
Now
And
Moreover the continuous function defined in the condition for completely regular space is said to be
separate the point x form the set A.
Tychonoff Space:
A completely regular T1-space is said to be a Tychonoff space or a
-space.
Note: It may be noted that since product of T1-space is T1-space and product of completely regular space is
completely regular space, so product of Tychonoff space is Tychonoff space.
Theorems:
o
o
o
o
o
o
Every completely regular space is a regular space as well.
Every completely regular T1-space is Hausdorff space or T2-space.
Every subspace of a completely regular space is completely regular space.
Product of completely regular space is a completely regular space.
Every subspace of Tychonoff space is Tychonoff space.
Every Tychonoff space is Hausdorff space.
Normal Space
Let X be a topological space and, A and B are disjoint closed subsets of X, then X is said to be
normal space, if there exist open sets U and V such that
,
.
In other words, a topological space X is said to be a normal space if for any disjoint pair of closed
sets F and G, there exist open sets U and V such that
,
.
Remarks:
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The collection of open sets separating the closed sets is called axiom-N.
It may be noted that some topologists consider the normal space basically T1 as well, while other do
not.
Every discrete space containing at least two elements in a normal space.
Every metric space is a normal space.
T4-Space: A normal T1-space is called T4-space.
Theorems:
o
o
o
Every closed subspace of normal space is normal space.
A closed continuous image of a normal space is normal.
A topological space is normal if and only if any closed set A and an open set Ucontaining A,
o
o
o
there is at least one open set V containing A such that
.
Every closed subspace of a T4-space is a T4-space.
Every T1 and normal space is a regular space.
If X is a normal space and f is closed continuous function from X onto a topological space Y,
then Y is normal as well.