topological properties and nets

topological properties and nets

\PMlinkescapephrase

locally compact Hausdorff space

Many topological properties and

concepts

can be translated in of

convergence

of nets. In this entry we give a list of this correspondence of properties. For detailed proofs please follow the .

1

Closure

Let

X

be a

topological space

and

Y

⊆

X

a subset. A point

x

∈

X

is in the

closure

of

Y

if and only there exists a net in

Y

converging (

http://planetmath.org/Net

) to

x

.

•

For a detailed proof, click here (

http://planetmath.org/NetsAndClosuresOfSubspaces

).

2

Closed

Let

X

be a topological space. A subset

Y

⊆

X

is closed if and only if every

convergent

net in

Y

converges

to a point in

Y

.

3

Limit point

Let

X

be a topological space and

Y

⊆

X

a subset. A point

x

∈

X

is a

limit point

of

Y

if and only if there is a net in

Y

converging to

x

that is not eventually constant.

4

Hausdorff

A topological space

X

is

Hausdorff

if and only if every convergent net in

X

has a unique limit.

5

Compact

A topological space

X

is

compact

if and only if every net in

X

has a convergent subnet.

•

For a detailed proof, click here (

http://planetmath.org/CompactnessAndConvergentSubnets

).

6

Continuous

Let

X

and

Y

be topological spaces. A

function

f

:

X

→

Y

is

continuous at

a point

x

∈

X

if and only if for every net

(

x

i

)

i

∈

I

in

X

converging to

x

, the net

(

f

⁢

(

x

i

)

)

i

∈

I

converges to

f

⁢

(

x

)

.

•

For a detailed proof, click here (

http://planetmath.org/ContinuityAndConvergentNets

).

7

Open map

Let

f

:

X

⟶

Y

be a surjective map between the topological spaces

X

and

Y

. Then

f

is an

open mapping

if and only if given a net

{

y

i

}

i

∈

I

⊂

Y

such that

y

i

⟶

y

, then for every

x

∈

f

-

1

⁢

(

{

y

}

)

there exists a subnet

{

y

i

j

}

j

∈

J

that to a net

{

x

i

j

}

j

∈

J

⊂

X

such that

x

i

j

⟶

x

. By ”” we that

{

x

i

j

}

j

∈

J

is such that

f

⁢

(

x

i

j

)

=

y

i

j

.

•

For a detailed proof, click here (

http://planetmath.org/ContinuousSurjectiveOpenMapsInTermsOfNets

).

8

Initial topology

Let

X

be a set,

{

X

α

}

α

∈

𝒜

a family of topological spaces and

f

α

:

X

→

X

α

a family of functions.

A net

(

x

i

)

i

∈

I

in

X

converges to a point

x

in the

initial topology

of

X

(relatively to the

mappings

f

α

) if and only if for each

α

∈

𝒜

,

(

f

α

⁢

(

x

i

)

)

i

∈

I

converges to

f

α

⁢

(

x

)

.

8.0.1

Particular case: subspace topology

Let

X

be a toplogical space and

Y

⊆

X

a subset. A net

(

y

i

)

i

∈

I

in

Y

converges to

y

∈

Y

in the

subspace topology

if and only if

(

y

i

)

i

∈

I

converges to

y

in the

topology

of

X

.

8.0.2

Particular case: product topology

Let

{

X

α

}

α

∈

𝒜

be a

collection

of topological spaces and

X

:=

∏

α

X

α

their

Cartesian product

. A net

(

x

i

)

i

∈

I

in

X

converges to

x

in the

product topology

if and only if every coordinate

(

x

i

α

)

i

∈

I

converges to

x

α

.

9

Compact-open topology

Let

X

be a locally compact Hausdorff space,

Y

a topological space and

C

⁢

(

X

,

Y

)

the set of

continuous functions

from

X

to

Y

. A net

(

f

i

)

i

∈

I

in

C

⁢

(

X

,

Y

)

converges to

f

in the

compact-open topology

if and only if whenever a net

(

x

i

)

i

∈

I

in

X

,

indexed by

the same

directed set

I

, converges to

x

∈

X

, we also have that

(

f

i

⁢

(

x

i

)

)

i

∈

I

converges to

f

⁢

(

x

)

.

Title

topological properties and nets

Canonical name

TopologicalPropertiesAndNets

Date of creation

2013-03-22 18:38:04

Last modified on

2013-03-22 18:38:04

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

6

Author

asteroid (17536)

Entry type

Feature

Classification

msc 54A20

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