Editing Total order (section)

===Order topology===

For any totally ordered set {{mvar|X}} we can define the ''[[interval (mathematics)|open interval]]s'' 
* {{math|1=(''a'', ''b'') = {{mset|''x'' | ''a'' < ''x'' and ''x'' < ''b''}}}}, 
* {{math|1=(−∞, ''b'') = {{mset|''x'' | ''x'' < ''b''}}}}, 
* {{math|1=(''a'', ∞) = {{mset|''x'' | ''a'' < ''x''}}}}, and 
* {{math|1=(−∞, ∞) = ''X''}}.  
We can use these open intervals to define a [[topology]] on any ordered set, the [[order topology]].

When more than one order is being used on a set one talks about the order topology induced by a particular order.  For instance if '''N''' is the natural numbers, {{char|<}} is less than and {{char|>}} greater than we might refer to the order topology on '''N''' induced by {{char|<}} and the order topology on '''N''' induced by {{char|>}} (in this case they happen to be identical but will not in general).

The order topology induced by a total order may be shown to be hereditarily [[Normal space|normal]].