eventual property

eventual property

Let

X

be a set and

P

a property on the elements of

X

. Let

(

x

i

)

i

∈

D

be a net (

D

a

directed set

) in

X

(that is,

x

i

∈

X

). As each

x

i

∈

X

,

x

i

either has or does not have property

P

. We say that the net

(

x

i

)

has property

P

above

j

∈

D

if

x

i

has property

P

for all

i

≥

j

. Furthermore, we say that

(

x

i

)

eventually

has property

P

if it has property

P

above some

j

∈

D

.

Examples

.

1.

Let

A

and

B

be non-empty sets. For

x

∈

A

, let

P

⁢

(

x

)

be the property that

x

∈

B

. So

P

is nothing more than the property of

elements

being in the

intersection

of

A

and

B

. A net

(

x

i

)

i

∈

D

eventually has

P

means that for some

j

∈

D

, the set

{

x

i

∣

i

∈

A

⁢

,

⁢

i

≥

j

}

⊆

B

. If

D

=

ℤ

, then we have that

A

and

B

eventually coincide.

2.

Now, suppose

A

is a

topological space

, and

B

is an open neighborhood of a point

x

∈

A

. For

y

∈

A

, let

P

B

⁢

(

y

)

be the property that

y

∈

B

. Then a net

(

x

i

)

has

P

B

eventually for every

neighborhood

B

of

x

is a

characterization

of convergence (to the point

x

, and

x

is the

accumulation point

of

(

x

i

)

).

3.

If

A

is a poset and

B

=

{

x

}

⊆

A

. For

y

∈

A

, let

P

⁢

(

y

)

again be the property that

y

=

x

. Let

(

x

i

)

be a net that eventually has property

P

. In other words,

(

x

i

)

is

eventually constant

. In particular, if for every chain

D

, the net

(

x

i

)

i

∈

D

is eventually constant in

A

, then we have a characterization of the

ascending chain condition

in

A

.

4.

directed net

. Let

R

be a

preorder

and let

(

x

i

)

i

∈

D

be a net in

R

. Let

x

⁢

(

D

)

be the image of the net:

x

⁢

(

D

)

=

{

x

i

∈

R

∣

i

∈

D

}

. Given a fixed

k

∈

D

and some

y

∈

x

⁢

(

D

)

, let

P

k

⁢

(

y

)

be the property (on

x

⁢

(

D

)

) that

x

k

≤

y

. Let

S

=

{

k

∈

D

∣

(

x

i

)

⁢

eventually has

⁢

P

k

}

.

If

S

=

D

, then we say that the net

(

x

i

)

is

directed

, or that

(

x

i

)

is a

directed net

. In other words, a directed net is a net

(

x

i

)

i

∈

D

such that for

every

i

∈

D

, there is a

k

⁢

(

i

)

∈

D

, such that

x

i

≤

x

j

for all

j

≥

k

⁢

(

i

)

.

If

(

x

i

)

i

∈

D

is a directed net, then

x

⁢

(

D

)

is a directed set: Pick

x

i

,

x

j

∈

x

⁢

(

D

)

, then there are

k

⁢

(

i

)

,

k

⁢

(

j

)

∈

D

such that

x

i

≤

x

m

for all

m

≥

k

⁢

(

i

)

and

x

j

≤

x

n

for all

n

≥

k

⁢

(

j

)

. Since

D

is directed, there is a

t

∈

D

such that

t

≥

k

⁢

(

i

)

and

t

≥

k

⁢

(

j

)

. So

x

t

≥

x

k

⁢

(

i

)

≥

x

i

and

x

t

≥

x

k

⁢

(

j

)

≥

x

j

.

However, if

(

x

i

)

i

∈

D

is a net such that

x

⁢

(

D

)

is directed,

(

x

i

)

need not be a directed net. For example, let

D

=

{

p

,

q

,

r

}

such that

p

≤

q

≤

r

, and

R

=

{

a

,

b

}

such that

a

≤

b

. Define a net

x

:

D

→

R

by

x

⁢

(

p

)

=

x

⁢

(

r

)

=

b

and

x

⁢

(

q

)

=

a

. Then

x

is not a directed net.

Remark

. The eventual property is a property on the class of nets (on a given set

X

and a given property

P

). We can write

Eventually

⁡

(

P

,

X

)

to denote its dependence on

X

and

P

.

Title

eventual property

Canonical name

EventualProperty

Date of creation

2013-03-22 16:34:45

Last modified on

2013-03-22 16:34:45

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

16

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06A06

Synonym

residually constant

Defines

eventually

Defines

directed net

Defines

eventually constant

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