ordinal space

ordinal space

Let

α

be an

ordinal

. The set

W

⁢

(

α

)

:=

{

β

∣

β

<

α

}

ordered by

≤

is a

well-ordered set

.

W

⁢

(

α

)

becomes a

topological space

if we equip

W

⁢

(

α

)

with the

interval topology

. An

ordinal space

X

is a topological space such that

X

=

W

⁢

(

α

)

(with the interval topology) for some ordinal

α

. In this entry, we will always assume that

W

⁢

(

α

)

≠

∅

, or

0

<

α

.

Before examining some basic topological structures of

W

⁢

(

α

)

, let us look at some of its order

structures

.

1.

First, it is easy to see that

W

(

α

)

=

↑

y

∪

W

(

y

)

, for any

y

∈

W

⁢

(

α

)

. Here,

↑

y

is the

upper set

of

y

.

2.

Another way of saying that

W

⁢

(

α

)

is well-ordered is that for any non-empyt subset

S

of

W

⁢

(

α

)

,

⋀

S

exists. Clearly,

0

∈

W

⁢

(

α

)

is its least

element

. If in

addition

1

<

α

,

W

⁢

(

α

)

is also atomic, with

1

as the sole atom.

3.

Next,

W

⁢

(

α

)

is

bounded complete

. If

S

⊆

W

⁢

(

α

)

is

bounded from above

by

a

∈

W

⁢

(

α

)

, then

b

=

⋁

S

is an ordinal such that

b

≤

a

<

α

, therefore

b

∈

W

⁢

(

α

)

as well.

4.

Finally, we note that

W

⁢

(

α

)

is a

complete lattice

iff

α

is not a

limit ordinal

. If

W

⁢

(

α

)

is

complete

, then

z

=

⋁

W

⁢

(

α

)

∈

W

⁢

(

α

)

. So

z

<

α

. This means that

z

+

1

≤

α

. If

z

+

1

<

α

, then

z

+

1

∈

W

⁢

(

α

)

so that

z

+

1

≤

⋁

W

⁢

(

α

)

=

z

, a

contradiction

. As a result,

z

+

1

=

α

. On the other hand, if

α

=

z

+

1

, then

z

=

⋁

W

⁢

(

α

)

∈

W

⁢

(

α

)

, so that

W

⁢

(

α

)

is complete.

In any ordinal space

W

⁢

(

α

)

where

0

<

α

, a typical

open interval

may be written

(

x

,

y

)

, where

0

≤

x

≤

y

<

α

. If

y

is not a limit ordinal, we can also write

(

x

,

y

)

=

[

x

+

1

,

z

]

where

z

+

1

=

y

. This means that

(

x

,

y

)

is a

clopen set

if

y

is not a limit ordinal. In particular, if

y

is not a limit ordinal, then

{

y

}

=

(

z

,

y

+

1

)

is clopen, where

z

+

1

=

y

, so that

y

is an

isolated point

. For example, any finite ordinal is an isolated point in

W

⁢

(

α

)

.

Conversely

, an isolated point can not be a limit ordinal. If

y

is isolated, then

{

y

}

is open. Write

{

y

}

as the union of open intervals

(

a

i

,

b

i

)

. So

a

i

<

y

<

b

i

. Since

y

+

1

covers

y

, each

b

i

must be

y

+

1

or

(

a

i

,

b

i

)

would contain more than a point. If

y

is a limit ordinal, then

a

i

<

a

i

+

1

<

y

so that, again,

(

a

i

,

b

i

)

would contain more than just

y

. Therefore,

y

can not be a limit ordinal and all

a

i

must be the same. Therefore

(

a

i

,

b

i

)

=

(

z

,

y

+

1

)

, where

z

is the

predecessor

of

y

:

z

+

1

=

y

.

Several basic properties of an ordinal space are:

1.

Isolated points in

W

⁢

(

α

)

are exactly those points that are limit ordinals (just a summary of the last two paragraphs).

2.

W

⁢

(

y

)

is open in

W

⁢

(

α

)

for any

y

∈

W

⁢

(

α

)

.

W

⁢

(

y

)

is closed iff

y

is not a limit ordinal.

3.

For any

y

∈

W

⁢

(

α

)

, the

collection

of intervals of the form

(

a

,

y

]

(where

a

<

y

) forms a

neighborhood base

of

y

.

4.

W

⁢

(

α

)

is a

normal space

for any

α

;

5.

W

⁢

(

α

)

is

compact

iff

α

is not a limit ordinal.

Some interesting ordinal spaces are

•

W

⁢

(

ω

)

, which is

homeomorphic

to the set of natural numbers

ℕ

.

•

W

⁢

(

ω

1

)

, where

ω

1

is the first uncountable ordinal.

W

⁢

(

ω

1

)

is often written

Ω

0

.

Ω

0

is not a

compact space

.

•

W

⁢

(

ω

1

+

1

)

, or

Ω

.

Ω

is compact, and, in fact, a

one-point compactification

of

Ω

0

.

References

1

S. Willard,

General Topology

, Addison-Wesley, Publishing Company, 1970.

Title

ordinal space

Canonical name

OrdinalSpace

Date of creation

2013-03-22 17:10:56

Last modified on

2013-03-22 17:10:56

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54F05

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