Order topology

In

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

${\displaystyle \{x\mid a<x\}}$

${\displaystyle \{x\mid x<b\}}$

for all a, b in X. Provided X has at least two elements, this is equivalent to saying that the open intervals

${\displaystyle (a,b)=\{x\mid a<x<b\}}$

together with the above rays form a base 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.

A

Let M = Z \ {−1} = (0,1), then M is connected, so M is dense on itself and has no gaps, in regards to <. If −1 is not the smallest or the largest element of Z, then ${\displaystyle (-\infty ,-1)}$ and ${\displaystyle (-1,\infty )}$ separate M, a contradiction. Assume without loss of generality that −1 is the smallest element of Z. Since {−1} is open in Z, there is some point p in M such that the interval (−1,p) is empty, so p is the minimum of M. Then M \ {p} = (0,p) ∪ (p,1) is not connected with respect to the subspace topology inherited from R. On the other hand, the subspace topology of M \ {p} inherited from the order topology of Z coincides with the order topology of M \ {p} induced by <, which is connected since there are no gaps in M \ {p} and it is dense. This is a contradiction.

Several variants of the order topology can be given:

• The right order topology^[2] on X is the topology having as a base all intervals of the form ${\displaystyle (a,\infty )=\{x\in X\mid x>a\}}$, together with the set X.
• The left order topology on X is the topology having as a base all intervals of the form ${\displaystyle (-\infty ,a)=\{x\in X\mid x<a\}}$, together with the set X.

These topologies naturally arise when working with

Additionally, these 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.

The left order topology is the standard topology used for many

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

${\displaystyle [0,\lambda )=\{\alpha \mid \alpha <\lambda \}}$

${\displaystyle [0,\lambda ]=\{\alpha \mid \alpha \leq \lambda \}}$

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; for finite ordinals, the order topology is simply the

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

Of particular interest is the case when λ = ω₁, the set of all countable ordinals, and the

• neither [0,ω₁) or [0,ω₁] is
  second-countable
• [0,ω₁] is
  paracompact

Any

The set of

The closed sets of a limit ordinal α are just the closed sets in the sense that we have already defined, namely, those that contain a limit ordinal whenever they contain all sufficiently large ordinals below it.

Any ordinal is, of course, an open subset of any larger ordinal. We can also define the topology on the ordinals in the following inductive way: 0 is the empty topological space, α+1 is obtained by taking the one-point compactification of α, and for δ a limit ordinal, δ is equipped with the inductive limit topology. Note that if α is a successor ordinal, then α is compact, in which case its one-point compactification α+1 is the disjoint union of α and a point.

As topological spaces, all the ordinals are

If α is a limit ordinal and X is a set, an α-indexed sequence of elements of X merely means a function from α to X. This concept, a transfinite sequence or ordinal-indexed sequence, is a generalization of the concept of a sequence. An ordinary sequence corresponds to the case α = ω.

If X is a topological space, we say that an α-indexed sequence of elements of X converges to a limit x when it converges as a

Ordinal-indexed sequences are more powerful than ordinary (ω-indexed) sequences to determine limits in topology: for example, ω₁ is a limit point of ω₁+1 (because it is a limit ordinal), and, indeed, it is the limit of the ω₁-indexed sequence which maps any ordinal less than ω₁ to itself: however, it is not the limit of any ordinary (ω-indexed) sequence in ω₁, since any such limit is less than or equal to the union of its elements, which is a countable union of countable sets, hence itself countable.

However, ordinal-indexed sequences are not powerful enough to replace nets (or filters) in general: for example, on the Tychonoff plank (the product space ${\displaystyle (\omega _{1}+1)\times (\omega +1)}$), the corner point ${\displaystyle (\omega _{1},\omega )}$ is a limit point (it is in the closure) of the open subset ${\displaystyle \omega _{1}\times \omega }$, but it is not the limit of an ordinal-indexed sequence.

• List of topologies
• Lower limit topology
• Long line (topology)
• Linear continuum
• Order topology (functional analysis)
• Partially ordered space