Order topology

In mathematics, the order topology is a topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets.

If X is a totally ordered set, the order topology on X is generated by the subbase of open rays

<math>(a, \infty) = \{ x \mid a < x\}<math>

<math>(-\infty, b) = \{x \mid x < b\}<math>

for some a,b in X. This is equivalent to saying that the open intervals

together with the above rays form a basis for the order topology. The open sets in X are the sets that are a union of (possibly infinitely many) such open intervals and rays.

The order topology makes X into a completely normal Hausdorff space.

The standard topologies on R, Q, and N are the order topologies.

Ordinal space

For any ordinal number λ one can consider the spaces of ordinal numbers

<math>[0,\lambda) = \{\alpha \mid \alpha < \lambda\}\,<math>

<math>[0,\lambda] = \{\alpha \mid \alpha \le \lambda\}\,<math>

together with the natural order topology. These spaces are called ordinal spaces. (Note that in the usual set-theoretic construction of ordinal numbers we have λ = [0,λ) and λ + 1 = [0,λ]). Obviously, these spaces are mostly of interest when λ is an infinite ordinal.

When λ = ω (the first infinite ordinal), the space [0,ω) is just N with the usual topology, while [0,ω] is the one-point compactification of N.

Of particular interest is the case when λ = ω₁, the first uncountable ordinal. The element ω₁ is a limit point of the subset [0,ω₁) even though no sequence of elements in [0,ω₁) has the element ω₁ as its limit. In particular, [0,ω₁] is not first-countable. The subspace [0,ω₁) is first-countable however, since the only point without a countable local base is ω₁. Some further properties include

neither [0,ω₁) or [0,ω₁] is separable or second-countable
[0,ω₁] is compact while [0,ω₁) is sequentially compact and countably compact, but not compact or paracompact

Left and right order topologies

Several interesting variants of the order topology can be given:

The left order topology on X is the topology whose open sets consist of intervals of the form (a, ∞).
The right order topology on X is the topology whose open sets consist of intervals of the form (−∞, b).

The left and right order topologies can be used to give counterexamples in general topology. For example, the left or right order topology on a bounded set provides an example of a compact space that is not Hausdorff.

Order topology

Ordinal space

Left and right order topologies

See also