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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" : : for all ''a,b'' in ''X''. 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. The order topology makes ''X'' into a completely normal Hausdorff space. The standard topologies on R, Q, and N are the order topologies. == Induced order topology == If ''Y'' is a subset of ''X'', then ''Y'' inherits a total order from ''X''. The set ''Y'' therefore has an order topology, the induced order topology. As a subset of ''X'', ''Y'' also has a subspace topology. The subspace topology is always at least as fine as the induced order topology, but they are not in general the same. For example, consider the subset ''Y'' = ∪ ''n''∈N in the rationals. Under the subspace topology, the singleton set is open in ''Y'', but under the induced order topology, any open set containing –1 must contain all but finitely many members of the space. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Order topology」の詳細全文を読む スポンサード リンク
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