derivation of properties of regular open set

derivation of properties of regular open set

Recall that a subset

A

of a

topological space

X

is

regular open

if it is equal to the

interior

of the

closure

of itself.

To facilitate further

analysis

of regular open sets, define the

operation

⊥

as follows:

A

⊥

:=

X

-

A

¯

.

Some of the

properties

of

⊥

and

regular

openness are listed and derived:

1.

For any

A

⊆

X

,

A

⊥

is open. This is

obvious

.

2.

⊥

reverses

inclusion

. This is also obvious.

3.

∅

⊥

=

X

and

X

⊥

=

∅

. This too is clear.

4.

A

∩

A

⊥

=

∅

, because

A

∩

A

⊥

⊆

A

∩

(

X

-

A

)

=

∅

.

5.

A

∪

A

⊥

is

dense in

X

, because

X

=

A

¯

∪

A

⊥

⊆

A

¯

∪

A

⊥

¯

=

A

∪

A

⊥

¯

.

6.

A

⊥

∪

B

⊥

⊆

(

A

∩

B

)

⊥

. To see this, first note that

A

∩

B

⊆

A

, so that

A

⊥

⊆

(

A

∩

B

)

⊥

. Similarly,

A

⊥

⊆

(

A

∩

B

)

⊥

. Take the union of the two inclusions and the result follows.

7.

A

⊥

∩

B

⊥

=

(

A

∪

B

)

⊥

. This can be verified by direct calculation:

A

⊥

∩

B

⊥

=

(

X

-

A

¯

)

∩

(

X

-

B

¯

)

=

X

-

(

A

¯

∪

B

¯

)

=

X

-

A

∪

B

¯

=

(

A

∪

B

)

⊥

.

8.

A

is regular open iff

A

=

A

⊥

⊥

. See the remark at the end of this entry (

http://planetmath.org/DerivationOfPropertiesOnInteriorOperation

).

9.

If

A

is open, then

A

⊥

is regular open.

Proof.

By the previous property, we want to show that

A

⊥

⁣

⊥

⊥

=

A

⊥

if

A

is open. For notational convenience, let us write

A

-

for the closure of

A

and

A

c

for the

complement

of

A

. As

=

-

c

⊥

, the equation now becomes

A

-

c

-

c

-

c

=

A

-

c

for any

open set

A

.

Since

A

⊆

A

-

for any set,

A

-

c

⊆

A

c

. This means

A

-

c

-

⊆

A

c

-

. Since

A

is open,

A

c

is closed, so that

A

c

-

=

A

c

. The last inclusion becomes

A

-

c

-

⊆

A

c

. Taking complement again, we have

A

⊆

A

-

c

-

c

.

(1)

Since

=

-

c

⊥

reverses inclusion, we have

A

-

c

-

c

-

c

⊆

A

-

c

, which is one of the inclusions. On the other hand, the inclusion (1) above applies to

any

open set, and because

A

-

c

is open,

A

-

c

⊆

A

-

c

-

c

-

c

, which is the other inclusion.
∎

10.

If

A

and

B

are regular open, then so is

A

∩

B

.

Proof.

Since

A

,

B

are regular open,

(

A

∩

B

)

⊥

⊥

=

(

A

⊥

⊥

∩

B

⊥

⊥

)

⊥

⊥

, which is equal to

(

A

⊥

∪

B

⊥

)

⊥

⁣

⊥

⊥

by property 7 above. Since

A

⊥

∪

B

⊥

is open, the last

expression

becomes

(

A

⊥

∪

B

⊥

)

⊥

by property 9, or

A

∩

B

by property 7 again.
∎

Remark

. All of the properties above can be dualized for regular closed sets. If fact, proving a property about regular closedness can be easily accomplished once we have the following:

(

*

)

A

is regular open iff

X

-

A

is regular closed.

Proof.

Suppose first that

A

is regular open. Then

int

⁡

(

X

-

A

)

¯

=

X

-

A

¯

¯

=

X

-

int

⁡

(

A

¯

)

=

X

-

A

. The

converse

is proved similarly.
∎

As a

corollary

, for example, we have: if

A

is closed, then

X

-

A

¯

is regular closed.

Proof.

If

A

is closed, then

X

-

A

is open, so that

(

X

-

A

)

⊥

=

X

-

X

-

A

¯

is regular open by property 9 above, which implies that

X

-

(

X

-

A

)

⊥

=

X

-

A

¯

is regular closed by

(

*

)

.
∎

Title

derivation of properties of regular open set

Canonical

name

DerivationOfPropertiesOfRegularOpenSet

Date of creation

2013-03-22 17:59:24

Last modified on

2013-03-22 17:59:24

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Derivation

Classification

msc 06E99

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