Editing Order topology (section)

== Induced order topology ==
If ''Y'' is a subset of ''X'', ''X'' a totally ordered set, 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 [[finer topology|fine]] as the induced order topology, but they are not in general the same.

For example, consider the subset ''Y'' = {−1} &cup; {1/''n''}<sub>''n''&isin;'''N'''</sub> of the [[rational number|rationals]].  Under the subspace topology, the [[singleton set]] {−1} 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.