Order Topology

An Order Topology is a Point-Set Topology that can be defined on any totally ordered set.

• Example(s):
  • Induced Order Topology,
  • Ordinal Number Topology.
  • a Real Line,
  • a Split Interval.
  • …
• Counter-Example(s)
  • Right Order Topology,
  • Left Order Topology,
  • Discrete Topology,
  • Topological Order.
• See: Orderable Space, Algebraic Topology, Differential Topology, Geometric Topology, Pointless Topology, Ordinal Space, Hausdorff Space, Mathematics, Topology, Totally Ordered Set, Real Numbers, Subbase, Interval (Mathematics), Base (Topology), Union (Set Theory), Topological Space, Completely Normal Space.

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References

2019

• (Wikipedia, 2019) ⇒ https://en.wikipedia.org/wiki/Order_topology Retrieved:2019-4-25.
  • In mathematics, an order topology is a certain 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]\displaystyle{ (a, \infty) = \{ x \mid a \lt x\} }[/math]

    [math]\displaystyle{ (-\infty, b) = \{x \mid x \lt b\} }[/math]

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

    [math]\displaystyle{ (a,b) = \{x \mid a \lt x \lt b\} }[/math]

    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 topological space X is called orderable if there exists a total order on its elements such that the order topology induced by that order and the given topology on X coincide. The order topology makes X into a completely normal Hausdorff space.

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

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