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ordinal space (Definition)

Let $ \alpha$ be an ordinal. The set $ W(\alpha):=\lbrace \beta \mid \beta < \alpha\rbrace$ ordered by $ \le$ is a well-ordered set. $ W(\alpha)$ becomes a topological space if we equip $ W(\alpha)$ with the interval topology. An ordinal space $ X$ is a topological space such that $ X=W(\alpha)$ (with the interval topology) for some ordinal $ \alpha$. In this entry, we will always assume that $ W(\alpha)\ne \varnothing$, or $ 0<\alpha$.

Before examining some basic topological structures of $ W(\alpha)$, let us look at some of its order structures.

  1. First, it is easy to see that $ W(\alpha)=\uparrow\!\!y \cup W(y)$, for any $ y\in W(\alpha)$. Here, $ \uparrow\!\!y$ is the upper set of $ y$.
  2. Another way of saying that $ W(\alpha)$ is well-ordered is that for any non-empyt subset $ S$ of $ W(\alpha)$, $ \bigwedge S$ exists. Clearly, $ 0\in W(\alpha)$ is its least element. If in addition $ 1<\alpha$, $ W(\alpha)$ is also atomic, with $ 1$ as the sole atom.
  3. Next, $ W(\alpha)$ is bounded complete. If $ S\subseteq W(\alpha)$ is bounded from above by $ a\in W(\alpha)$, then $ b=\bigvee S$ is an ordinal such that $ b\le a<\alpha$, therefore $ b\in W(\alpha)$ as well.
  4. Finally, we note that $ W(\alpha)$ is a complete lattice iff $ \alpha$ is not a limit ordinal. If $ W(\alpha)$ is complete, then $ z=\bigvee W(\alpha)\in W(\alpha)$. So $ z<\alpha$. This means that $ z+1\le \alpha$. If $ z+1<\alpha$, then $ z+1\in W(\alpha)$ so that $ z+1\le \bigvee W(\alpha)=z$, a contradiction. As a result, $ z+1=\alpha$. On the other hand, if $ \alpha=z+1$, then $ z=\bigvee W(\alpha)\in W(\alpha)$, so that $ W(\alpha)$ is complete.

In any ordinal space $ W(\alpha)$ where $ 0<\alpha$, a typical open interval may be written $ (x,y)$, where $ 0\le x\le y<\alpha$. 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 $ \lbrace y\rbrace = (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(\alpha)$.

Conversely, an isolated point can not be a limit ordinal. If $ y$ is isolated, then $ \lbrace y\rbrace$ is open. Write $ \lbrace y\rbrace$ 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(\alpha)$ are exactly those points that are limit ordinals (just a summary of the last two paragraphs).
  2. $ W(y)$ is open in $ W(\alpha)$ for any $ y\in W(\alpha)$. $ W(y)$ is closed iff $ y$ is not a limit ordinal.
  3. For any $ y\in W(\alpha)$, the collection of intervals of the form $ (a,y]$ (where $ a<y$) forms a neighborhood base of $ y$.
  4. $ W(\alpha)$ is a normal space for any $ \alpha$;
  5. $ W(\alpha)$ is compact iff $ \alpha$ is not a limit ordinal.

Some interesting ordinal spaces are

Bibliography

1
S. Willard, General Topology, Addison-Wesley, Publishing Company, 1970.



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Cross-references: one-point compactification, uncountable, natural numbers, homeomorphic, compact, normal space, neighborhood base, intervals, collection, closed, properties, point, contain, covers, union, open, isolated, finite, isolated point, clopen set, open interval, contradiction, complete, limit ordinal, iff, complete lattice, bounded from above, bounded complete, atom, addition, least element, subset, upper set, easy to see, order, structures, interval topology, topological space, well-ordered set, ordinal
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This is version 5 of ordinal space, born on 2007-06-01, modified 2007-06-02.
Object id is 9498, canonical name is OrdinalSpace.
Accessed 743 times total.

Classification:
AMS MSC: 54F05 (General topology :: Special properties :: Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces)
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