ordered space

ordered space

Definition

. A set

X

that is both a

topological space

and a poset is variously called a

topological ordered space

,

ordered topological space

, or simply an

ordered space

. Note that there is no compatibility conditions imposed on

X

. In other words, the

topology

𝒯

and the

partial ordering

≤

on

X

operate independently of one another.

If the

partial order

is a

total order

, then

X

is called a

totally ordered space

. In some literature, a totally ordered space is called an ordered space. In this entry, however, an ordered space is always a

partially

ordered space.

One can construct an ordered space from a set with fewer

structures

.

1.

For example, any topological space is trivially an ordered space, with the partial order defined by

a

≤

b

iff

a

=

b

. But this is not so interesting. A more interesting example is to take a

T

0

space

X

, and define

a

≤

b

iff

a

∈

{

b

}

¯

. The

relation

so defined turns out to be a partial order on

X

, called the

specialization order

, making

X

an ordered space.

2.

On the other hand, given any poset

P

, we can arbitrarily assign a topology on it, making it an ordered space, so that every poset is trivially an ordered space. Again this is not very interesting.

3.

A slightly more useful example is to take a poset

P

, and take

ℒ

⁢

(

P

)

:=

{

P

-

↑

x

∣

x

∈

P

}

,

the family of all set

complements

of principal

upper sets

of

P

, as the

subbasis

for the topology

ω

⁢

(

P

)

of

P

. The topology

ω

⁢

(

P

)

so generated is called the

lower topology

on

P

.

4.

Dually, if we take

𝒰

⁢

(

P

)

:=

{

P

-

↓

x

∣

x

∈

P

}

,

as the subbasis, we get the

upper topology

on

P

, denoted by

ν

⁢

(

P

)

.

5.

In the lower topology

ω

⁢

(

P

)

of

P

, if

y

∈

P

-

↑

x

, then either

y

<

x

(

strict inequality

) or

x

∥

y

(

incomparable

with

x

). If

x

is an

isolated

element

, then

P

-

↑

x

=

P

-

{

x

}

. This means that

{

x

}

is a

closed set

. Similarly,

{

x

}

is closed in the upper topology

ν

⁢

(

P

)

.

If

x

is the top element of

P

, then

{

x

}

is a closed set in

ω

⁢

(

P

)

, since

P

-

↑

x

=

P

-

{

x

}

is open. Similarly

{

x

}

is closed in

ν

⁢

(

P

)

if

x

is the bottom element in

P

.

If

P

is

totally ordered

, there are no isolated elements. As a result, we may write

P

-

↑

x

in a more familiar fashion:

(

-

∞

,

x

)

. Similarly,

P

-

↓

x

may be written as

(

x

,

∞

)

.

6.

Things get more interesting when we take the

common refinement

of

ω

⁢

(

P

)

and

ν

⁢

(

P

)

. What we end up with is called the

interval topology

of

P

.

When

P

is totally ordered, the interval topology on

P

has

ℐ

⁢

(

P

)

:=

{

(

x

,

y

)

∣

x

,

y

∈

P

}

as a subbasis, where

(

x

,

y

)

denotes the

open

poset interval

, consisting of elements

a

∈

P

such that

x

<

a

<

y

. Since finite

intersections

of open poset intervals is a poset interval, an

open set

in

P

can be written as an (arbitrary) union of open poset intervals.

As an example, the

usual topology

on

ℝ

is precisely the interval topology generated by the linear order on

ℝ

.

Remark

. It is a common practice in mathematics to impose special compatibility conditions on a structure having two inherent

substructures

so the substructures inter-relate, so that one can derive more interesting fruitful results. This is true also in the case of an ordered space. Let

X

be an ordered space. Below are some of the common conditions that can be imposed on

X

:

•

X

is said to be

upper semiclosed

if

↑

x

is a closed set for every

x

∈

X

.

•

Similarly,

X

is

lower semiclosed

if

↓

x

is closed in

X

.

•

X

is

semiclosed

if it is both upper and lower semiclosed.

•

If

≤

, as a subset of

X

×

X

, is closed in the

product topology

, then

X

is called a

pospace

.

Other structures, such as

ordered topological vector spaces

, topological lattices (

http://planetmath.org/TopologicalLattice

), and topological vector lattices (

http://planetmath.org/TopologicalVectorLattice

) are ordered spaces with

algebraic structures

satisfying certain additional compatibility conditions. Please click on the links for details.

References

1

G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, D. S. Scott,

Continuous

Lattices and Domains

, Cambridge University Press, Cambridge (2003).

Title

ordered space

Canonical name

OrderedSpace

Date of creation

2013-03-22 17:05:36

Last modified on

2013-03-22 17:05:36

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

15

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54E99

Classification

msc 06F20

Classification

msc 06F30

Synonym

ordered topological space

Synonym

topological ordered space

Synonym

partially ordered space

Synonym

partially ordered topological space

Related topic

OrderTopology

Defines

upper topology

Defines

lower topology

Defines

interval topology

Defines

upper semiclosed

Defines

lower semiclosed

Defines

semiclosed

Defines

pospace

Defines

totally ordered space

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