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ordinal space∗
CWoo†
2013-03-21 23:04:45
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) 6= ∅, 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
V 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 W
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.
W
If W (α) is complete, then z = W (α) ∈ W (α). So z < α.WThis 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,
W
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).
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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 (ai , bi ). So ai < y < bi .
Since y + 1 covers y, each bi must be y + 1 or (ai , bi ) would contain more than a
point. If y is a limit ordinal, then ai < ai + 1 < y so that, again, (ai , bi ) would
contain more than just y. Therefore, y can not be a limit ordinal and all ai
must be the same. Therefore (ai , bi ) = (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 (ω1 ), where ω1 is the first uncountable ordinal. W (ω1 ) is often written
Ω0 . Ω0 is not a compact space.
• W (ω1 + 1), or Ω. Ω is compact, and, in fact, a one-point compactification
of Ω0 .
References
[1] S. Willard, General Topology, Addison-Wesley, Publishing Company, 1970.
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