Kuratowski closure-complement theorem

Kuratowski closure-complement theorem

Problem

. Let

X

be a

topological space

and

A

a subset of

X

. How many (distinct) sets can be obtained by iteratively applying the

closure

and

complement

operations

to

A

?

Kuratowski studied this problem, and showed that at most

14

sets that can be generated from a given set in an arbitrary topological space. This is known as the

Kuratowski closure-complement theorem

.

Let us examine this problem more closely. For convenience, let us denote

:

-

X

→

X

be the

closure operator

:

A

↦

A

-

,

and

:

c

X

→

X

the

complementation

operator

:

A

↦

A

c

.

A set that can be obtained from

A

by iteratively applying

-

and

c

has the form

A

σ

, where

σ

is an operator on

X

that is the

composition

of finitely many

-

and

c

. In other words,

σ

is a word on the

alphabet

{

-

,

c

}

.

First, notice that

A

-

-

=

A

-

and

A

c

⁢

c

=

A

. This means that

σ

can be

reduced

(or simplified) to a form such that

-

and

c

occurs alternately.

In

addition

, we have the following:

Proposition 1

.

A

-

c

-

c

-

c

-

=

A

-

c

-

.

Proof.

For any set

A

in a topological space

X

,

A

-

is closed, so that

A

-

c

-

is regular closed. This means that

A

-

c

-

=

A

-

c

-

c

-

c

-

.
∎

This means that

σ

can be reduced to one of the following cases:

1

,

-

,

-

c

,

-

c

-

,

-

c

-

c

,

-

c

-

c

-

,

-

c

-

c

-

c

,

c

,

c

-

,

c

-

c

,

c

-

c

-

,

c

-

c

-

c

,

c

-

c

-

c

-

,

c

-

c

-

c

-

c

,

where

1

=

c

⁢

c

is the

identity operator

. As there are a total of 14

combinations

, proving the closure-complement

theorem

is to exhibit an example. To do this, pick

X

=

ℝ

, the

real line

. Let

A

=

(

0

,

1

)

∪

{

2

}

∪

(

(

3

,

4

)

∩

ℚ

)

∪

(

(

5

,

7

)

-

{

6

}

)

. In other words,

A

is the union of a real

interval

, a point, a

rational

interval, and a real interval with a point deleted. Then

1.

A

-

=

[

0

,

1

]

∪

{

2

}

∪

[

3

,

4

]

∪

[

5

,

7

]

,

2.

A

-

c

=

(

-

∞

,

0

)

∪

(

1

,

2

)

∪

(

2

,

3

)

∪

(

4

,

5

)

∪

(

7

,

∞

)

,

3.

A

-

c

-

=

(

-

∞

,

0

]

∪

[

1

,

3

]

∪

[

4

,

5

]

∪

[

7

,

∞

)

,

4.

A

-

c

-

c

=

(

0

,

1

)

∪

(

3

,

4

)

∪

(

5

,

7

)

,

5.

A

-

c

-

c

-

=

[

0

,

1

]

∪

[

3

,

4

]

∪

[

5

,

7

]

,

6.

A

-

c

-

c

-

c

=

(

-

∞

,

0

)

∪

(

1

,

3

)

∪

(

4

,

5

)

∪

(

7

,

∞

)

,

7.

A

c

=

(

-

∞

,

0

]

∪

[

1

,

2

)

∪

(

2

,

3

]

∪

(

(

3

,

4

)

-

ℚ

)

∪

[

4

,

5

]

∪

{

6

}

∪

[

7

,

∞

)

,

8.

A

c

-

=

(

-

∞

,

0

]

∪

[

1

,

5

]

∪

{

6

}

∪

[

7

,

∞

)

,

9.

A

c

-

c

=

(

0

,

1

)

∪

(

5

,

6

)

∪

(

6

,

7

)

,

10.

A

c

-

c

-

=

[

0

,

1

]

∪

[

5

,

7

]

,

11.

A

c

-

c

-

c

=

(

-

∞

,

0

)

∪

(

1

,

5

)

∪

(

7

,

∞

)

,

12.

A

c

-

c

-

c

-

=

(

-

∞

,

0

]

∪

[

1

,

5

]

∪

[

7

,

∞

)

,

13.

A

c

-

c

-

c

-

c

=

(

0

,

1

)

∪

(

5

,

7

)

,

together with

A

, are

14

pairwise distinct sets that can be generated by

-

and

c

.

Title

Kuratowski closure-complement theorem

Canonical name

KuratowskiClosurecomplementTheorem

Date of creation

2013-03-22 17:59:28

Last modified on

2013-03-22 17:59:28

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 54A99

Classification

msc 54A05

Generated on Sat Feb 10 11:15:53 2018 by

LaTeXML