ordinal space

Let α be an ordinalMathworldPlanetmathPlanetmath. The set W(α):={ββ<α} ordered by is a well-ordered set. W(α) becomes a topological spaceMathworldPlanetmath 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 structuresMathworldPlanetmath.

  1. 1.

    First, it is easy to see that W(α)=yW(y), for any yW(α). Here, y is the upper set of y.

  2. 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, 0W(α) is its least element. If in addition 1<α, W(α) is also atomic, with 1 as the sole atom.

  3. 3.

    Next, W(α) is bounded complete. If SW(α) is bounded from above by aW(α), then b=S is an ordinal such that ba<α, therefore bW(α) as well.

  4. 4.

    Finally, we note that W(α) is a complete latticeMathworldPlanetmath iff α is not a limit ordinalMathworldPlanetmath. If W(α) is completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, then z=W(α)W(α). So z<α. This means that z+1α. If z+1<α, then z+1W(α) so that z+1W(α)=z, a contradictionMathworldPlanetmathPlanetmath. 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 0xy<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 (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. 1.

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

  2. 2.

    W(y) is open in W(α) for any yW(α). W(y) is closed iff y is not a limit ordinal.

  3. 3.

    For any yW(α), the collectionMathworldPlanetmath of intervals of the form (a,y] (where a<y) forms a neighborhood base of y.

  4. 4.

    W(α) is a normal spaceMathworldPlanetmath for any α;

  5. 5.

    W(α) is compactPlanetmathPlanetmath iff α is not a limit ordinal.

Some interesting ordinal spaces are

  • W(ω), which is homeomorphic to the set of natural numbers .

  • 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.


  • 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