If X is a totally ordered set, and a and b are elements of X, we can define the open intervals (a, b) = {x : a < x and x < b}, (−∞, b) = {x : x < b}, (a, ∞) = {x : a < x} and (−∞, ∞) = X. The order topology on X consists all sets that are a union of (possibly infinitely many) such open intervals. The order topology makes X into a normal Hausdorff space. The open intervals form a base for the order topology.
Several interesting variants of the order topology can be given:
The left order topology on X is the topology whose open sets consist of intervals of the form (a, ∞).
The right order topology on X is the topology whose open sets consist of intervals of the form (−∞, b).
The left and right order 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.
In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space.
The trivial topology is the topology with the least possible number of open sets, since the definition of a topology requires these two sets to be open.
The trivial topology belongs to a pseudometric space in which the distance between any two points is zero, and to a uniform space in which the whole cartesian product X × X is the only entourage.
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