topological lattice

topological lattice

A

topological lattice

is a

lattice

L

equipped with a

topology

𝒯

such that the meet and join

operations

from

L

×

L

(with the

product topology

) to

L

are

continuous

.

Let

(

x

i

)

i

∈

I

be a net in

L

. We say that

(

x

i

)

converges

to

x

∈

L

if

(

x

i

)

is

eventually

in any open neighborhood of

x

, and we write

x

i

→

x

.

Remarks

•

If

(

x

i

)

and

(

y

j

)

are nets,

indexed by

I

,

J

respectively, then

(

x

i

∧

y

j

)

and

(

x

i

∨

y

j

)

are nets, both indexed by

I

×

J

. This is clear, and is stated in preparation for the

proposition

below.

•

If

x

i

→

x

and

y

j

→

y

, then

x

i

∧

y

j

→

x

∧

y

and

x

i

∨

y

j

→

x

∨

y

.

Proof.

Let’s show the first convergence, and the other one follows similarly. The

function

f

:

x

↦

(

x

,

y

)

↦

x

∧

y

is a

continuous function

, being the composition of two continuous functions. If

x

∧

y

∈

U

is open, then

x

∈

f

-

1

⁢

(

U

)

is open. As

x

i

→

x

, there is an

i

0

∈

I

such that

x

i

∈

f

-

1

⁢

(

U

)

for all

i

≥

i

0

, which means that

x

i

∧

y

=

f

⁢

(

x

i

)

∈

U

. By the same token, for each

i

∈

I

, the function

g

i

:

y

↦

(

x

i

,

y

)

↦

x

i

∧

y

is a continuous function. Since

x

i

∧

y

∈

U

is open,

y

∈

g

-

1

⁢

(

U

)

is open. As

y

j

→

y

, there is a

j

0

∈

J

such that

y

j

∈

g

-

1

⁢

(

U

)

for all

j

≥

j

0

, or

x

i

∧

y

j

=

g

i

⁢

(

y

j

)

∈

U

, for all

i

≥

i

0

and

j

≥

j

0

. Hence

x

i

∧

y

j

→

x

∧

y

.
∎

•

For any net

(

x

i

)

, the set

A

=

{

a

∈

L

∣

x

i

→

a

}

is a sublattice of

L

.

Proof.

If

a

,

b

∈

A

, then

x

i

=

x

i

∧

x

i

→

a

∧

b

. So

a

∧

b

∈

A

. Similarly

a

∨

b

∈

A

.
∎

There are two approaches to finding examples of topological lattices. One way is to start with a

topological space

X

such that

X

is partially ordered, then find two continuous binary operations on

X

to form the meet and join operations of a lattice. The

real numbers

ℝ

, with operations defined by

a

∧

b

=

inf

⁡

{

a

,

b

}

and

a

∨

b

=

sup

⁡

{

a

,

b

}

, is one such an example. This can be easily generalized to the space of real-valued continuous functions, since, given any two real-valued continuous functions

f

and

g

,

f

∨

g

:=

max

⁡

(

f

,

g

)

⁢

and

⁢

f

∧

g

:=

min

⁡

(

f

,

g

)

are well-defined real-valued continuous functions as well (in fact, it is enough to say that for any continuous function

f

, its

absolute value

|

f

|

is also continuous, so that

max

⁡

(

f

,

0

)

=

1

2

⁢

(

f

+

|

f

|

)

,

and thus

max

⁡

(

f

,

g

)

=

max

⁡

(

f

-

g

,

0

)

+

g

⁢

and

⁢

min

⁡

(

f

,

g

)

=

f

+

g

-

max

⁡

(

f

,

g

)

are both continuous as well).

The second approach is to start with a general lattice

L

and define a topology

𝒯

on the subsets of the underlying set of

L

, with the hope that both

∨

and

∧

are continuous under

𝒯

. The obvious example using this second approach is to take the

discrete topology

of the underlying set. Another way is to impose conditions, such as requiring that the lattice be meet and

join continuous

. Of course, finding a topology on the underlying set of a lattice may not guarantee a topological lattice.

Title

topological lattice

Canonical name

TopologicalLattice

Date of creation

2013-03-22 15:47:26

Last modified on

2013-03-22 15:47:26

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

19

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06F30

Classification

msc 06B30

Classification

msc 54H12

Related topic

OrderedVectorSpace

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