Scott topology

Scott topology

Let

P

be a dcpo. A subset

U

of

P

is said to be

Scott open

if it

satisfies

the following two conditions:

1.

U

an

upper set

:

↑

U

=

U

, and

2.

if

D

is a

directed set

with

⋁

D

∈

U

, then there is a

y

∈

D

such that

(

↑

y

)

∩

D

⊆

U

.

Condition 2 is

equivalent

to saying that

U

has non-empty

intersection

with

D

whenever

D

is directed and its

supremum

is in

U

.

For example, for any

x

∈

P

, the set

U

⁢

(

x

)

:=

P

-

(

↓

x

)

is Scott open: if

y

∈

↑

U

(

x

)

, then there is

z

∈

U

⁢

(

x

)

with

z

≤

y

. Since

z

∉

↓

x

,

y

∉

↓

x

. So

y

∈

U

⁢

(

x

)

, or that

U

⁢

(

x

)

is upper. If

D

is directed and

e

≤

x

for all

e

∈

D

, then

d

:=

⋁

D

≤

x

as well. Therefore,

d

∈

U

⁢

(

x

)

implies

e

∈

U

⁢

(

x

)

for some

e

∈

D

. Hence

U

⁢

(

x

)

is Scott open.

The

collection

σ

⁢

(

P

)

of all Scott open sets of

P

is a

topology

, called the

Scott topology

of

P

, named after its inventor Dana Scott. Let us prove that

σ

⁢

(

P

)

is indeed a topology:

Proof.

We verify each of the axioms of an

open set

:

•

Clearly

P

itself is Scott open, and

∅

is

vacuously

Scott open.

•

Suppose

U

and

V

are Scott open. Let

W

=

U

∩

V

and

b

∈

↑

W

. Then for some

a

∈

W

,

a

≤

b

. Since

a

∈

U

∩

V

,

b

∈

↑

U

=

U

and

b

∈

↑

V

=

V

. This means

b

∈

W

, so

W

is an upper set. Next, if

D

is directed with

⋁

D

∈

W

, then,

⋁

D

∈

U

∩

V

. So there are

y

,

z

∈

D

with

(

↑

y

)

∩

D

⊆

U

and

(

↑

z

)

∩

D

⊆

V

. Since

D

is directed, there is

t

∈

D

such that

t

∈

(

↑

y

)

∩

(

↑

z

)

. So

(

↑

t

)

∩

D

⊆

(

↑

y

)

∩

(

↑

z

)

∩

D

=

(

(

↑

y

)

∩

D

)

∩

(

(

↑

z

)

∩

D

)

⊆

U

∩

V

=

W

. This means that

W

is Scott open.

•

Suppose

U

i

are open and

i

∈

I

an

index set

. Let

U

=

⋃

{

U

i

∣

i

∈

I

}

and

b

∈

↑

U

. So

a

≤

b

for some

a

∈

U

. Since

a

∈

U

i

for some

i

∈

I

,

b

∈

↑

U

i

=

U

i

as

U

i

is upper. Hence

b

∈

U

i

⊆

U

, or that

U

is upper. Next, suppose

D

is directed with

⋁

D

∈

U

. Then

⋁

D

∈

U

i

for some

i

∈

I

. Since

U

i

is Scott open, there is

y

∈

D

with

(

↑

y

)

∩

D

⊆

U

i

⊆

U

, so

U

is Scott open.

Since the Scott open sets satisfy the axioms of a topology,

σ

⁢

(

P

)

is a topology on

P

.
∎

Examples

. If

P

is the unit interval:

P

=

[

0

,

1

]

, then

P

is a

complete chain

, hence a dcpo. Any Scott open set has the form

(

a

,

1

]

if

0

<

a

≤

1

, or

[

0

,

1

]

. If

P

=

[

0

,

1

]

×

[

0

,

1

]

, the unit square, then

P

is a dcpo as it is already a

continuous lattice

. The Scott open sets of

P

are any upper subset of

P

that is also an open set in the usual sense.

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

Scott topology

Canonical

name

ScottTopology

Date of creation

2013-03-22 16:49:29

Last modified on

2013-03-22 16:49:29

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06B35

Defines

Scott open

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