open set

open set

In a

metric space

M

a set

O

is called an

open subset

of

M

or just

open

, if for every

x

∈

O

there is an open ball

S

around

x

such that

S

⊂

O

. If

d

⁢

(

x

,

y

)

is the

distance

from

x

to

y

then the open ball

B

r

with radius

r

>

0

around

x

is given as:

B

r

=

{

y

∈

M

|

d

⁢

(

x

,

y

)

<

r

}

.

Using the idea of an open ball one can define a

neighborhood

of a point

x

. A set containing

x

is called a neighborhood of

x

if there is an open ball around

x

which is a subset of the neighborhood.

These neighborhoods have some properties, which can be used to define a

topological space

using the

Hausdorff axioms

for neighborhoods, by which again an open set within a topological space can be defined. In this way we drop the metric and get the more general topological space. We can define a topological space

X

with a set of neighborhoods of

x

called

U

x

for every

x

∈

X

, which

satisfy

1.

x

∈

U

for every

U

∈

U

x

2.

If

U

∈

U

x

and

V

⊂

X

and

U

⊂

V

then

V

∈

U

x

(every set containing a neighborhood of

x

is a neighborhood of

x

itself).

3.

If

U

,

V

∈

U

x

then

U

∩

V

∈

U

x

.

4.

For every

U

∈

U

x

there is a

V

∈

U

x

, such that

V

⊂

U

and

V

∈

U

p

for every

p

∈

V

.

The last point leads us back to open sets, indeed a set

O

is called open if it is a neighborhood of every of its points. Using the properties of these open sets we arrive at the usual definition of a topological space using open sets, which is

equivalent

to the above definition. In this definition we look at a set

X

and a set of subsets of

X

, which we call open sets, called

𝒪

, having the following properties:

1.

∅

∈

𝒪

and

X

∈

𝒪

.

2.

Any union of open sets is open.

3.

intersections

of open sets are open.

Note that a topological space is more general than a metric space, i.e. on every metric space a

topology

can be defined using the open sets from the metric, yet we cannot always define a metric on a topological space such that all open sets remain open.

Examples:

•

On the real axis the

interval

I

=

(

0

,

1

)

is open because for every

a

∈

I

the open ball with radius

min

⁡

(

a

,

1

-

a

)

is always a subset of

I

. (Using the

standard metric

d

⁢

(

x

,

y

)

=

|

x

-

y

|

.)

•

The open ball

B

r

around

x

is open. Indeed, for every

y

∈

B

r

the open ball with radius

r

-

d

⁢

(

x

,

y

)

around y is a subset of

B

r

, because for every

z

within this ball we have:

d

⁢

(

x

,

z

)

≤

d

⁢

(

x

,

y

)

+

d

⁢

(

y

,

z

)

<

d

⁢

(

x

,

y

)

+

r

-

d

⁢

(

x

,

y

)

=

r

.

So

d

⁢

(

x

,

z

)

<

r

and thus

z

is in

B

r

. This holds for every

z

in the ball around

y

and therefore it is a subset of

B

r

•

A non-metric topology would be the finite

complement

topology on

infinite sets

, in which a set is called open, if its complement is finite.

Title

open set

Canonical name

OpenSet

Date of creation

2013-03-22 12:39:25

Last modified on

2013-03-22 12:39:25

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

21

Author

mathwizard (128)

Entry type

Definition

Classification

msc 54A05

Synonym

open

Synonym

open subset

Defines

Hausdorff axioms

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