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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). ∗ hOrdinalSpacei created: h2013-03-21i by: hCWooi version: h39498i Privacy setting: h1i hDefinitioni h54F05i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1 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. 2