# order topology

Let $(X,\le )$ 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

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

$$ |

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

$$ |

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 $$. If there is a $z$ such that $$, then $(-\mathrm{\infty},z)$ and $(z,\mathrm{\infty})$ are disjoint open sets separating $x$ and $y$. If no $z$ were between $x$ and $y$, then $(-\mathrm{\infty},y)$ and $(x,\mathrm{\infty})$ 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 |