Ordinal Spaces and Initial Segments

Ordinals as Ordered Spaces

Ordinal intervals give concrete examples of ordered spaces whose points need not behave like real numbers. Their order has a first point, may have points with an immediate predecessor or successor, and may also have limit points in the order itself. To describe the resulting topology, we first specify the ordinal order and the intervals on which it is used.

We use the standard set-theoretic definition: an ordinal is a transitive set well-ordered by the membership relation \(\in\). Here, transitive means that if \(y\) belongs to the set and \(x\in y\), then \(x\) belongs to the set as well; well-ordered means that every nonempty subset has a least element. Ordinals are compared by the ordinal order: \(\beta<\gamma\) means \(\beta\in\gamma\), and \(\beta\leq\gamma\) means \(\beta<\gamma\) or \(\beta=\gamma\). In the ordinal order, \(\beta<\gamma\) is equivalent to \(\beta\) being a proper subset of \(\gamma\).

For example, the finite ordinals \(0,1,2,3\) agree with the corresponding nonnegative integers. Each positive finite ordinal is the successor of the preceding one. The ordinal \(\omega\), in contrast, has no immediate predecessor: every finite ordinal \(n\) is followed by \(n+1<\omega\). This is the distinction between a successor ordinal and a limit ordinal at the level of the order.

Ordinal Intervals and Initial Segments

For ordinals \(\beta\leq\gamma\), write \([\beta,\gamma]=\{\xi:\beta\leq\xi\leq\gamma\}\), and use the corresponding strict or half-open interval notation when an endpoint is excluded. In particular, \([0,\alpha]=\{\xi:\xi\leq\alpha\}=\alpha+1\), while \([0,\alpha)=\{\xi:\xi<\alpha\}=\alpha\). The interval \([0,\alpha]\) is an ordered set with the ordinal order restricted to its elements.

Proof. If \(S\) is an initial segment, then \(S\) is transitive: if \(y\in S\) and \(x\in y\), then \(x<y\), so the initial-segment condition gives \(x\in S\). The membership order on \(S\) is a well-order because it is the restriction of the well-order on \(\alpha\). Thus \(S\) is itself an ordinal. Set \(\beta=S\). Since \(S\subseteq\alpha\), the ordinal order agrees with inclusion and gives \(\beta\leq\alpha\). The elements of the ordinal \(\beta=S\) are exactly the ordinals less than \(\beta\), so \(S=[0,\beta)\). For uniqueness, if \([0,\beta)=[0,\gamma)\), then these sets are the ordinals \(\beta\) and \(\gamma\) themselves, so \(\beta=\gamma\). \(\square\)

This result explains why initial segments of an ordinal do not have arbitrary shapes: once an element is included, every smaller element must also be included. It also makes the notation useful in examples. The initial segment ending just before \(\omega\) consists of all finite ordinals; the segment ending just before \(\omega+1\) includes \(\omega\) as well.

The Order Topology on an Ordinal Interval

Give \([0,\alpha]\) its order topology: the topology generated by open intervals and open rays in its linear order. A basis for this topology consists of the whole interval together with sets of the following forms, whenever the indicated endpoints lie in the interval:

Proof. In the order topology, basic sets before restricting to \([0,\alpha]\) are open intervals and open rays. An open interval with endpoints \(\beta<\gamma\) restricts to \((\beta,\gamma)\). A ray consisting of elements less than \(\gamma\) restricts to \([0,\gamma)\), because \(0\) is the least element of the interval. A ray consisting of elements greater than \(\beta\) restricts to \((\beta,\alpha]\), because \(\alpha\) is its greatest element. The whole interval is also open. These restrictions therefore give the order-topology basis on \([0,\alpha]\), including at its endpoints. \(\square\)

The notation \((\beta,\gamma)\) always means the elements strictly between the stated ordinal endpoints, not the real numbers between them. In particular, an interval can be empty if its endpoints are adjacent. The topology is determined by the order: it is not obtained by treating ordinal labels as real coordinates.

Common Pitfall: Losing Track of the Endpoint

A frequent error is to treat \([0,\omega)\) and \([0,\omega]\) as the same set. They are not: the first excludes \(\omega\), while the second includes it. In particular, \([0,\omega)=\{0,1,2,\ldots\}\), and \([0,\omega]=\{0,1,2,\ldots,\omega\}\).

Another tempting but incorrect calculation is to identify \(\omega+1\) with the finite ordinal \(1\), or to write \([0,\omega+1]=\{0,1,2,\ldots\}\). The successor definition gives \(\omega+1=\omega\cup\{\omega\}\), so the correct interval is \(\{0,1,2,\ldots,\omega,\omega+1\}\). The endpoint \(\omega+1\) is a new point after all finite ordinals and after \(\omega\). Keeping the order and endpoint conventions explicit prevents both errors.

Ordinal intervals are useful because they make order topology concrete: one can list points, identify initial segments, and compute basis elements directly from the order. Finite intervals show how adjacent points can produce open singletons; infinite intervals show that their basic sets must be read from the ordinal order rather than from the familiar geometry of the real line.