continuous relation

continuous relation

The idea of a

continuous relation

is neither as old nor as well-established as the idea of a

continuous function

.
Different authors use somewhat different

definitions

. The present article is based on the following definition:

Let

X

and

Y

be

topological spaces

and

R

a

relation

between

X

and

Y

(

R

is a subset of

X

×

Y

).

R

is said to be if

for any

open subset

V

of

Y

,

R

-

1

⁢

(

V

)

is open in

X

.

Here

R

-

1

⁢

(

V

)

is the

inverse image

of

V

under

R

, and is defined as

R

-

1

⁢

(

V

)

:=

{

x

∈

X

∣

x

⁢

R

⁢

y

⁢

for some

⁢

y

∈

V

}

.

Equivalently,

R

is a continuous relation if for any open set

V

of

Y

, the set

π

X

⁢

(

(

X

×

V

)

∩

R

)

is open in

X

, where

π

X

is the

projection map

X

×

Y

→

X

.

Remark

. Continuous relations are

generalization

of continuous functions: if a continuous relation is also a

function

, then it is a continuous function.

Some examples.

•

Let

X

be an

ordered space

. Then the

partial order

≤

is

continuous

iff for every open subset

A

of

X

, its

lower set

↓

A

is also open in

X

.

In particular, in

ℝ

, the usual

linear ordering

≤

on

ℝ

is continuous. To see this, let

A

be an open subset of

ℝ

. If

A

=

∅

, then

↓

A

=

∅

as well, and so is open. Suppose now that

A

is non-empty and deal with the case when

A

is not

bounded from above

. If

r

∈

ℝ

, then there is

a

∈

A

such that

r

≤

a

, so that

r

∈

↓

A

, which implies

↓

A

=

ℝ

. Hence

↓

A

is open. If

A

is bounded from above, then

A

has a

supremum

(since

ℝ

is

Dedekind complete

), say

x

. Since

A

is open,

x

∉

A

(or else

x

∈

(

a

,

b

)

⊆

A

, implying

x

<

x

+

b

2

∈

(

a

,

b

)

, contradicting the fact that

x

is the least

upper bound

of

A

). So

↓

A

=

(

-

∞

,

x

)

, which is open also. Therefore,

≤

is a continuous relation on

ℝ

.

•

Again, we look at the space

ℝ

with its usual interval topology. The relation this time is

R

=

{

(

x

,

y

)

∈

ℝ

2

∣

x

2

+

y

2

=

1

}

. This is not a continuous relation. Take

A

=

(

-

2

,

2

)

, which is open. But then

R

-

1

⁢

(

A

)

=

[

-

1

,

1

]

, which is closed.

•

Now, let

X

be a

locally connected

topological space. For any

x

,

y

∈

X

, define

x

∼

y

iff

x

and

y

belong to the same

connected component

of

X

. Let

A

be an open subset of

X

. Then

B

=

∼

-

1

(

A

)

is the union of all connected components containing points of

A

. Since (it can be shown) each connected component is open, so is their union, and hence

B

is open. Thus

∼

is a continuous relation.

•

If

R

is

symmetric

, then

R

is continuous iff

R

-

1

is. In particular, in a topological space

X

, an

equivalence relation

∼

on

X

is continuous iff the

projection

p

of

X

onto the

quotient space

X

/

∼

is an

open mapping

.

Remark

. Alternative definitions:
One apparently common definition (as described by Wyler) is to require inverse images of open sets to be open and inverse
images of

closed sets

to be closed (making the relation upper and

lower semi-continuous

). Wyler suggests the following
definition:
If

r

:

e

→

f

is a relation between topological spaces

E

and

F

, then

r

is continuous iff for each topological space

A

, and functions

f

:

A

→

E

and

g

:

A

→

F

such that

f

⁢

(

u

)

𝑟

g

⁢

(

u

)

for all

u

∈

A

,

continuity

of

f

implies continuity of

g

.

References

1

T. S. Blyth,

Lattices and Ordered

Algebraic Structures

, Springer, New York (2005).

2

J. L. Kelley,

General Topology

, D. van Nostrand Company, Inc., 1955.

3

Oswald Wyler,

http://links.jstor.org/sici?sici=0002-9939%28197108%2929%3A3%3C588%3AACORIT%3E2.0.CO%3B2-3

A

Characterization

of

Regularity

in

Topology

Proceedings of the American Mathematical Society, Vol. 29, No. 3. (Aug., 1971), pp. 588-590.

Title

continuous relation

Canonical

name

ContinuousRelation

Date of creation

2013-03-22 17:05:39

Last modified on

2013-03-22 17:05:39

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54A99

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