# Order Topology

### (General Topology)

Tech

###### General Topology ## What is Order topology?

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

### Technology Types

definite quantityfunctional analysisgeneral topologyorder theoryordinal numbertopological space

### Synonyms

Induced order topologyLeft order topologyOrder topologyOrderable topological spaceOrdinal spaceRight order topology

## Tech Info

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— Date merged: 11/6/2021, 1:32:43 PM
— Date scraped: 5/20/2021, 6:18:54 PM