Let be a linearly ordered set. The order topology on is defined to be the topology generated by the subbasis consisting of open rays, that is sets of the form

This is equivalent to saying that is generated by the basis of open intervals; that is, the open rays as defined above, together with sets of the form

The standard topologies on , and are the same as the order topologies on these sets.

If is a subset of , then is a linearly ordered set under the induced order from . Therefore, has an order topology defined by this ordering, the induced order topology. Moreover, has a subspace topology which it inherits as a subspace of the topological space . The subspace topology is always finer than the induced order topology, but they are not in general the same.

For example, consider the subset . Under the subspace topology, the singleton set is open in , but under the order topology on , any open set containing must contain all but finitely many members of the space.

A chain under the order topology is Hausdorff: pick any two distinct points ; without loss of generality, say . If there is a such that , then and are disjoint open sets separating and . If no were between and , then and are disjoint open sets separating and .