# order topology

Let $(X,\leq)$ be a linearly ordered set. The order topology on $X$ is defined to be the topology $\mathcal{T}$ generated by the subbasis consisting of open rays, that is sets of the form

 $(x,\infty)=\{y\in X|y>x\}$
 $(-\infty,x)=\{y\in X|y

for some $x\in X$.

This is equivalent to saying that $\mathcal{T}$ is generated by the basis of open intervals; that is, the open rays as defined above, together with sets of the form

 $(x,y)=\{z\in X|x

for some $x,y\in X$.

The standard topologies on $\mathbb{R}$, $\mathbb{Q}$ and $\mathbb{N}$ are the same as the order topologies on these sets.

If $Y$ is a subset of $X$, then $Y$ is a linearly ordered set under the induced order from $X$. Therefore, $Y$ has an order topology $\mathcal{S}$ defined by this ordering, the induced order topology. Moreover, $Y$ has a subspace topology $\mathcal{T}^{\prime}$ which it inherits as a subspace of the topological space $X$. The subspace topology is always finer than the induced order topology, but they are not in general the same.

For example, consider the subset $Y=\{-1\}\cup\{\frac{1}{n}\mid n\in\mathbb{N}\}\subseteq\mathbb{Q}$. Under the subspace topology, the singleton set $\{-1\}$ is open in $Y$, but under the order topology on $Y$, any open set containing $-1$ must contain all but finitely many members of the space.

A chain $X$ under the order topology is Hausdorff: pick any two distinct points $x,y\in X$; without loss of generality, say $x. If there is a $z$ such that $x, then $(-\infty,z)$ and $(z,\infty)$ are disjoint open sets separating $x$ and $y$. If no $z$ were between $x$ and $y$, then $(-\infty,y)$ and $(x,\infty)$ are disjoint open sets separating $x$ and $y$.

 Title order topology Canonical name OrderTopology Date of creation 2013-03-22 12:10:34 Last modified on 2013-03-22 12:10:34 Owner rspuzio (6075) Last modified by rspuzio (6075) Numerical id 10 Author rspuzio (6075) Entry type Definition Classification msc 54B99 Classification msc 06F30 Synonym induced order topology Related topic OrderedSpace Related topic LinearContinuum Related topic ProofOfGeneralizedIntermediateValueTheorem Related topic ASpaceIsConnectedUnderTheOrderedTopologyIfAndOnlyIfItIsALinearContinuum