locale

locale

Topology

, in its most abstract sense, is the study of a family of subsets, called

open sets

, of some given set

X

, when subject to certain conditions based purely on set-theoretic

operations

. Namely, these conditions are that the

intersection

of two open sets is an open set, union of open sets is an open set, and that the

empty set

and

X

are themselves open sets. One can thus think of a topology as a certain kind of a

lattice

L

⁢

(

X

)

of subsets associated with the set

X

, and the study of a general

topological space

can be distilled further, to the study of this particular lattice

L

⁢

(

X

)

. In this setting, the basic elements under scrutiny are no longer “points” in

X

, but elements of

L

⁢

(

X

)

. This shift in focus is the starting point of the so-called “pointless topology”, where the study of topology takes on a lattice-theoretic flavor.

Specifically, the lattice

L

⁢

(

X

)

is an example of what is known as a locale. Formally, a

locale

is a

complete lattice

L

that is meet infinitely

distributive

:

x

∧

(

⋁

Y

)

=

⋁

(

x

∧

Y

)

for any

Y

⊆

L

, where

x

∧

Y

:=

{

x

∧

y

∣

y

∈

Y

}

.

A “pointless” proof

. To see that the lattice

L

⁢

(

X

)

of open sets of a topological space

X

is a locale, we first observe that

L

⁢

(

X

)

is a complete lattice, since arbitrary joins of open sets are open by definition, which is enough to ensure that arbitrary meets exist too (although they are not arbitrary intersections, they are

interiors

of the intersections). This means that

⋁

Y

and

⋁

(

x

⁢

⋀

Y

)

are both open sets, and hence the expressions in the

equality

are at least meaningful at this point. Further,

⋁

(

x

∧

Y

)

=

⋃

(

x

∩

Y

)

⊆

x

∩

(

⋃

Y

)

=

x

∧

(

⋁

Y

)

, because

x

∩

y

⊆

x

∩

(

⋃

Y

)

for

individual

y

∈

Y

. So we are left with showing that

x

∧

(

⋁

Y

)

⊆

⋁

(

x

∧

Y

)

. Again, this is true at an individual level:

x

∩

y

⊆

⋃

(

x

∩

Y

)

.

(1)

Let’s write

z

=

⋃

(

x

∩

Y

)

. Then the expression above becomes

y

∩

x

⊆

z

, where

x

,

y

,

z

are all open sets. Put it another way,

y

=

(

y

∩

x

)

∪

(

y

-

x

)

(2)

⊆

z

∪

(

y

-

x

)

(3)

=

z

∪

(

y

∩

x

c

)

(4)

=

(

z

∪

y

)

∩

(

z

∪

x

c

)

(5)

⊆

(

z

∪

y

)

∩

(

z

∪

x

c

)

∘

,

(6)

where

-

denotes the

set difference

operator

,

c

is the set

complementation

, and

∘

the

interior operator

. (6) comes from the fact that

y

is open. Now, take the union of all

y

∈

Y

, and write this union

t

:=

⋃

Y

=

⋃

{

y

∣

y

∈

Y

}

. Then

t

⊆

(

z

∪

t

)

∩

(

z

∪

x

c

)

∘

(7)

⊆

(

z

∪

t

)

∩

(

z

∪

x

c

)

(8)

=

z

∪

(

t

∩

x

c

)

.

(9)

Taking the intersection with

x

on both sides, we have

t

∩

x

⊆

(

z

∪

(

t

∩

x

c

)

)

∩

x

(10)

=

(

z

∩

x

)

∪

(

(

t

-

x

)

∩

x

)

(11)

=

(

z

∩

x

)

∪

∅

(12)

=

z

∩

x

(13)

⊆

z

.

(14)

Substituting

t

and

z

back to their original form, we have

x

∩

(

⋃

Y

)

⊆

⋃

(

x

∩

Y

)

, which is what we wanted to prove.

Notice that in the proof above, no points of

X

are employed, and everything is done via basic set operations, as well as extra set operations, such as the interior operator.

Remarks

.

•

Another thing worthy of note is the following fact:

A lattice is a locale iff it is a

complete

Brouwerian lattice

(equivalently a

complete Heyting algebra

).

A sketch of proof goes as follows: if

L

is a locale, then for any

a

,

b

∈

L

, the

element

⋁

{

c

∣

a

∧

c

≤

b

}

is the relative pseudocomplement of

a

in

b

.

Conversely

, if

L

is a

Heyting algebra

, then we can use the trick that for every

y

∈

Y

,

y

≤

x

→

⋁

(

x

∧

Y

)

, to show

x

∧

(

⋁

Y

)

≤

⋁

(

x

∧

Y

)

.

•

Other examples of locales found in topology is the lattice of

regular open sets

in a topological space

X

.

•

Complete Boolean algebras

also provide examples of locales. But a locale need not be complete

Boolean

. The

interval topology

on the real line is such an example.

•

A

locale homomorphism

ϕ

is a

lattice homomorphism

between two locales such that arbitrary sups are preserved:

ϕ

⁢

(

⋁

i

∈

I

{

a

i

}

)

=

⋁

i

∈

I

{

ϕ

⁢

(

a

i

)

}

.

•

A locale is also called a

frame

, and a frame homomorphism a locale homomorphism. However, P. T. Johnstone distinguishes the two names when they are considered as

categories

. The category

Fra

of frames consists of locales (or frames) as objects, and locale homomorphisms as

morphisms

. On the other hand, the category

Loc

of locales is defined as the opposite category of

Fra

.

•

In

[

5

]

, a frame and a locale are two distinct objects. A frame is defined as above. But a locale

L

is a triple

(

F

,

P

,

⊧

)

, where

F

is a frame,

P

is the set of frame homomorphisms from

F

to

{

0

,

1

}

, the

trivial

frame (= the trivial

Boolean algebra

), and

⊧

is a subset (

relation

) of

P

×

F

given by

f

⊧

a

iff

f

⁢

(

a

)

=

1

. For each

a

∈

F

, let

U

⁢

(

a

)

=

{

f

∈

P

∣

f

⊧

a

}

. It is not hard to see that the

collection

of all sets of the form

U

⁢

(

a

)

forms a topology on

P

. As a result, every frame can be viewed as a topology on some set!

References

1

F. Borceux,

Handbook of Categorical Algebra 3: Categories of Sheaves

, Cambridge University Press (1994).

2

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).

3

P. T. Johnstone,

The Art of Pointless Thinking: A Student’s Guide to the Category of Locales,

Category Theory

at Work (Bremen, 1990)

, pp 85-107, Res. Exp. Math., 18, Heldermann, Berlin (1991).

4

P. T. Johnstone,

Stone Spaces

, Cambridge University Press, Cambridge (1982).

5

S. Vickers,

Topology via Logic

, Cambridge University Press, Cambridge (1989).

Title

locale

Canonical name

Locale

Date of creation

2013-03-22 16:38:11

Last modified on

2013-03-22 16:38:11

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06D22

Synonym

frame

Synonym

frame homomorphism

Related topic

CompleteHeytingAlgebra

Defines

locale homomorphism

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