derivation of properties on interior operation

derivation of properties on interior operation

Let

X

be a

topological space

and

A

a subset of

X

. Then

1.

int

⁡

(

A

)

⊆

A

.

Proof.

If

a

∈

int

⁡

(

A

)

, then

a

∈

U

for some

open set

U

⊆

A

. So

a

∈

A

.
∎

2.

int

⁡

(

A

)

is open.

Proof.

Since

int

⁡

(

A

)

is a union of open sets,

int

⁡

(

A

)

is open.
∎

3.

int

⁡

(

A

)

is the largest open set

contained

in

A

.

Proof.

If

U

is open set with

int

⁡

(

A

)

⊆

U

⊆

A

, then

U

⊆

⋃

{

V

⊆

A

∣

V

⁢

open

}

=

int

⁡

(

A

)

, so

U

=

int

⁡

(

A

)

.
∎

4.

A

is open if and only if

A

=

int

⁡

(

A

)

.

Proof.

If

A

is open, then

A

is the largest open set contained in

A

, and so

int

⁡

(

A

)

=

A

by

property

3 above. On the other hand, if

int

⁡

(

A

)

=

A

, then

A

is open, since

int

⁡

(

A

)

is, by property 2 above.
∎

5.

int

⁡

(

int

⁡

(

A

)

)

=

int

⁡

(

A

)

.

Proof.

Since

int

⁡

(

A

)

is open by property 2,

int

⁡

(

A

)

=

int

⁡

(

int

⁡

(

A

)

)

by property 4.
∎

6.

int

⁡

(

X

)

=

X

and

int

⁡

(

∅

)

=

∅

.

Proof.

This is so because both

X

and

∅

are open sets.
∎

7.

A

∁

¯

=

(

int

⁡

(

A

)

)

∁

.

Proof.

(LHS

⊆

RHS). If

a

∈

A

∁

¯

, then

a

∈

B

for every

closed set

B

such that

A

∁

⊆

B

. In particular,

a

∈

(

int

⁡

(

A

)

)

∁

, for

(

int

⁡

(

A

)

)

∁

is the

complement

of an open set by property 2, and

A

∁

⊆

(

int

⁡

(

A

)

)

∁

by taking the complement of property 1.

(RHS

⊆

LHS). If

a

∈

(

int

⁡

(

A

)

)

∁

, then

a

∉

int

⁡

(

A

)

. If

B

is a closed set such that

A

∁

⊆

B

, then

B

∁

⊆

A

. Since

B

∁

is open,

B

∁

⊆

int

⁡

(

A

)

by property 3, so

a

∉

B

∁

, and thus

a

∈

B

. Since

B

is arbitrary,

a

∈

A

∁

¯

as desired.
∎

8.

A

¯

∁

=

int

⁡

(

A

∁

)

.

Proof.

Set

B

=

A

∁

, and apply property 7. So

A

¯

∁

=

B

∁

¯

∁

=

(

int

⁡

(

B

)

)

∁

⁢

∁

=

int

⁡

(

B

)

=

int

⁡

(

A

∁

)

.
∎

9.

A

⊆

B

implies that

int

⁡

(

A

)

⊆

int

⁡

(

B

)

.

Proof.

This is so because

int

⁡

(

A

)

is open (property 2), contained in

A

(and therefore contained in

B

), so contained in

int

⁡

(

B

)

, as

int

⁡

(

B

)

is the largest open set contained in

B

(property 3).
∎

10.

int

⁡

(

A

)

=

A

∖

∂

⁡

A

, where

∂

⁡

A

is the

boundary

of

A

.

Proof.

Recall that

∂

⁡

A

=

A

¯

∩

A

∁

¯

. So

∂

⁡

A

=

A

¯

∩

(

int

⁡

(

A

)

)

∁

by property 7. By direct computation, we have

A

∖

∂

⁡

A

=

A

∖

(

A

¯

∩

(

int

⁡

(

A

)

)

∁

)

=

(

A

∖

A

¯

)

∪

(

A

∖

(

int

⁡

(

A

)

)

∁

)

. Since

A

∖

A

¯

=

∅

and

A

∖

(

int

⁡

(

A

)

)

∁

=

A

∩

(

int

⁡

(

A

)

)

∁

⁢

∁

=

A

∩

int

⁡

(

A

)

, which is

int

⁡

(

A

)

by property 2.
∎

11.

A

¯

=

int

⁡

(

A

)

∪

∂

⁡

A

.

Proof.

Again, by direct computation:

int

⁡

(

A

)

∪

∂

⁡

A

=

int

⁡

(

A

)

∪

(

A

¯

∩

(

int

⁡

(

A

)

)

∁

)

⁢

because

⁢

∂

⁡

A

=

A

¯

∩

(

int

⁡

(

A

)

)

∁

=

(

int

⁡

(

A

)

∪

A

¯

)

∩

(

int

⁡

(

A

)

∪

(

int

⁡

(

A

)

)

∁

)

∩

distributes over

∪

=

A

¯

∩

X

=

A

¯

.

⁢

int

⁡

(

A

)

⊆

A

⊆

A

¯

∎

12.

X

=

int

⁡

(

A

)

∪

∂

⁡

A

∪

int

⁡

(

A

∁

)

.

Proof.

By property 11,

int

⁡

(

A

)

∪

∂

⁡

A

∪

int

⁡

(

A

∁

)

=

A

¯

∪

int

⁡

(

A

∁

)

, which, by property 8, is

A

¯

∪

A

¯

∁

, and the last

expression

is just

X

.
∎

13.

int

⁡

(

A

∩

B

)

=

int

⁡

(

A

)

∩

int

⁡

(

B

)

.

Proof.

(LHS

⊆

RHS). Let

C

=

int

⁡

(

A

∩

B

)

. Since

C

is open and contained in both

A

and

B

,

C

is contained in both

int

⁡

(

A

)

and

int

⁡

(

B

)

, since

int

⁡

(

A

)

and

int

⁡

(

B

)

are the largest open sets in

A

and

B

respectively. (RHS

⊆

LHS). Let

D

=

int

⁡

(

A

)

∩

int

⁡

(

B

)

. So

D

is open and is a subset of both

A

and

B

, hence a subset of

A

∩

B

, and therefore a subset of

int

⁡

(

A

∩

B

)

, since it is the largest open set contained in

A

∩

B

.
∎

Remark

. Using property 7, we see that an alternative

definition

of

interior

can be given:

int

⁡

(

A

)

=

A

∁

¯

∁

.

Title

derivation of properties on interior operation

Canonical

name

DerivationOfPropertiesOnInteriorOperation

Date of creation

2013-03-22 17:55:28

Last modified on

2013-03-22 17:55:28

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Derivation

Classification

msc 54-00

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