Ordinal space | Wikipedia audio article
Channel: wikipedia tts
Date: 17/05/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Order_topology


00:01:32 1 Induced order topology
00:02:13 2 An example of a subspace of a linearly ordered space whose topology is not an order topology
00:02:37 3 Left and right order topologies
00:02:55 4 Ordinal space
00:05:40 5 Topology and ordinals
00:05:51 5.1 Ordinals as topological spaces
00:07:01 5.2 Ordinal-indexed sequences
00:07:40 6 See also
00:09:26 7 Notes
00:09:37 8 References
00:13:36 Ordinal-indexed sequences
00:15:49 See also



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SUMMARY
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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"

  
    
      
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    {\displaystyle (a,\infty )=\{x\mid ax\}}
  

  
    
      
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    {\displaystyle (-\infty ,b)=\{x\mid xb\}}
  for all a,b in X. This is equivalent to saying that the open intervals

  
    
      
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    {\displaystyle (a,b)=\{x\mid axb\}}
  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.