separation axioms

separation axioms

The

separation axioms

are additional conditions which may be required to a

topological space

in order to ensure that some particular types of sets can be

separated

by

open sets

, thus avoiding certain pathological cases.

Axiom

Definition

T

0

given two distinct points, there is an open set containing exactly one of them;

T

1

(

http://planetmath.org/T1Space

)

given two distinct points, there is a

neighborhood

of each of them which does not contain the other point;

T

2

(

http://planetmath.org/T2Space

)

given two distinct points, there are two

disjoint

open sets each of which contains one of the points;

T

2

⁤

1

2

given two distinct points, there are two open sets, each of which contains one of the points, whose

closures

are disjoint;

T

3

(

http://planetmath.org/T3Space

)

given a

closed set

A

and a point

x

∉

A

, there are two disjoint open sets

U

and

V

such that

x

∈

U

and

A

⊂

V

;

T

3

⁤

1

2

given a closed set

A

and a point

x

∉

A

, there is an

Urysohn function

for

A

and

{

b

}

;

T

4

given two disjoint closed sets

A

and

B

, there are two disjoint open sets

U

and

V

such that

A

⊂

U

and

B

⊂

V

;

T

5

given two separated sets

A

and

B

, there are two disjoint open sets

U

and

V

such that

A

⊂

U

and

B

⊂

V

.

If a topological space

satisfies

a

T

i

axiom, it is called a

T

i

-space.
The following table shows other common names for topological spaces with these or other additional separation properties.

Name

Separation properties

Kolmogorov space

T

0

Fréchet space

T

1

Hausdorff space

T

2

Completely Hausdorff space

T

2

⁤

1

2

Regular space

T

3

and

T

0

Tychonoff

or completely regular space

T

3

⁤

1

2

and

T

0

Normal space

T

4

and

T

1

Perfectly

T

4

space

T

4

and every closed set is a

G

δ

(see here (

http://planetmath.org/G_deltaSet

))

Perfectly normal space

T

1

and perfectly

T

4

Completely normal

space

T

5

and

T

1

The following

implications

hold strictly:

(

T

2

⁢

and

⁢

T

3

)

⇒

T

2

⁤

1

2

(

T

3

⁢

and

⁢

T

4

)

⇒

T

3

⁤

1

2

T

3

⁤

1

2

⇒

T

3

T

5

⇒

T

4

Completely normal

⇒

normal

⇒

completely regular

⇒

regular

⇒

T

2

⁤

1

2

⇒

T

2

⇒

T

1

⇒

T

0

Remark.

Some authors define

T

3

spaces in the way we defined regular spaces, and

T

4

spaces in the way we defined normal spaces (and vice-versa); there is no consensus on this issue.

Bibliography:

Counterexamples in Topology

, L. A. Steen, J. A. Seebach Jr., Dover Publications Inc. (New York)

Title

separation axioms

Canonical name

SeparationAxioms

Date of creation

2013-03-22 13:28:47

Last modified on

2013-03-22 13:28:47

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

26

Author

Koro (127)

Entry type

Definition

Classification

msc 54D10

Classification

msc 54D15

Synonym

separation properties

Related topic

NormalTopologicalSpace

Related topic

HausdorffSpaceNotCompletelyHausdorff

Related topic

SierpinskiSpace

Related topic

MetricSpacesAreHausdorff

Related topic

ZeroDimensional

Related topic

T2Space

Related topic

RegularSpace

Related topic

T4Space

Defines

Hausdorff

Defines

completely Hausdorff

Defines

normal

Defines

completely normal

Defines

regular

Defines

Tychonoff

Defines

completely regular

Defines

perfectly normal

Defines

Tychonov

Defines

perfectly

T

4

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