for some .
for some .
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 .
|Date of creation||2013-03-22 12:10:34|
|Last modified on||2013-03-22 12:10:34|
|Last modified by||rspuzio (6075)|
|Synonym||induced order topology|