topology via converging nets

topology via converging nets

Given a

topological space

X

, one can define the

concept

of

convergence

of a

sequence

, and more generally, the convergence of a net.

Conversely

, given a set

X

, a class of nets, and a suitable definition of “convergence” of a net, we can topologize

X

. The procedure is done as follows:

Let

C

be the class of all pairs of the form

(

x

,

y

)

where

x

is a net in

X

and

y

is an element of

X

. For any subset

U

of

X

with

y

∈

U

, we say that a net

x

converges

to

y

with respect to

U

if

x

is eventually in

U

. We denote this by

x

→

U

y

. Let

𝒯

:=

{

U

⊆

X

∣

(

x

,

y

)

∈

C

⁢

and

⁢

y

∈

U

⁢

imply

⁢

x

→

U

y

}

.

Then

𝒯

is a

topology

on

X

.

Proof.

Clearly

x

→

X

y

for any pair

(

x

,

y

)

∈

C

. In

addition

,

x

→

∅

y

is

vacuously true

. For any

U

,

V

∈

𝒯

, we want to show that

W

:=

U

∩

V

∈

𝒯

. Since

x

is eventually in

U

and

V

, there are

i

,

j

∈

D

(where

D

is the domain of

x

), such that

x

r

∈

U

and

x

s

∈

V

for all

r

≥

i

and

s

≥

j

. Since

D

is directed, there is a

k

∈

D

such that

k

≥

i

and

k

≥

j

. It is clear that

x

k

∈

W

and that any

t

≥

k

we have that

x

t

∈

W

as well. Next, if

U

α

are sets in

𝒯

, we want to show their union

U

:=

⋃

{

U

α

}

is also in

𝒯

. If

y

is a point in

U

then

y

is a point in some

U

α

. Since

(

x

,

y

)

∈

C

with

x

is eventually in

U

α

, we have that

x

is eventually in

U

as well.
∎

Remark

. The above can be generalized. In fact, if the class of pairs

(

x

,

y

)

satisfies

some “axioms” that are commonly found as properties of convergence, then

X

can be topologized. Specifically, let

X

be a set and

C

again be the class of all pairs

(

x

,

y

)

as described above. A

subclass

𝒞

of

C

is called a

convergence class

if the following conditions are satisfied

1.

x

is a

constant

net with value

y

∈

X

, then

(

x

,

y

)

∈

𝒞

2.

(

x

,

y

)

∈

𝒞

implies

(

z

,

y

)

∈

𝒞

for any subnet

z

of

x

3.

if every subnet

z

of a net

x

has a subnet

t

with

(

t

,

y

)

∈

𝒞

, then

(

x

,

y

)

∈

𝒞

4.

suppose

(

x

,

y

)

∈

𝒞

with

D

=

dom

⁡

(

x

)

, and for each

i

∈

D

, we have that

(

z

i

,

x

i

)

∈

𝒞

, with

D

i

=

dom

⁡

(

z

i

)

. Then

(

z

,

x

)

∈

𝒞

, where

z

is the net whose domain is

D

×

F

with

F

:=

∏

{

D

i

∣

i

∈

D

}

, given by

z

⁢

(

i

,

f

)

=

(

i

,

f

⁢

(

i

)

)

.

If

(

x

,

y

)

∈

𝒞

, we write

x

→

y

or

lim

D

⁡

x

=

y

. The last condition can then be visualized as

⋮

⋮

⋮

⋮

⋮

⋱

⋯

z

i

⁢

a

⋮

z

j

⁢

f

⋮

z

k

⁢

p

⋯

z

i

⁢

f

⁢

(

i

)

⋯

⋮

⋮

⋮

⋮

⋮

⋯

⋱

⋯

z

i

⁢

b

⋮

z

j

⁢

g

⋮

z

k

⁢

q

⋯

z

j

⁢

f

⁢

(

j

)

⋯

⋮

⋮

⋮

⋮

⋮

⋯

⇒

⋱

⋯

z

i

⁢

c

⋮

z

j

⁢

h

⋮

z

k

⁢

r

⋯

z

k

⁢

f

⁢

(

k

)

⋯

⋮

⋮

⋮

⋮

⋮

⋯

⋱

⋯

↓

⋮

↓

⋮

↓

⋯

↘

⋯

x

i

⋯

x

j

⋯

x

k

⋯

→

y

y

,

which is reminiscent of Cantor’s diagonal argument.

Now, for any subset

A

of

X

, we define

A

c

to be the subset of

X

consisting of all points

y

∈

X

such that there is a net

x

in

A

with

x

→

y

. It can be shown that

c

is a

closure operator

, which induces a topology

𝒯

𝒞

on

X

. Furthermore, under this induced topology, the notion of converging nets (as defined by the topology) is exactly the same as the notion of convergence described by the convergence class

𝒞

.

In addition, it may be shown that there is a

one-to-one correspondence

between the topologies and the convergence classes on the set

X

. The correspondence is order reversing in the sense that if

𝒞

1

⊆

𝒞

2

as

convergent

classes, then

𝒯

𝒞

2

⊆

𝒯

𝒞

1

as topologies.

Title

topology via converging nets

Canonical name

TopologyViaConvergingNets

Date of creation

2013-03-22 17:14:27

Last modified on

2013-03-22 17:14:27

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54A20

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