Order topology

Definition

Suppose $X$ is a linearly ordered set (i.e., a set equipped with a total ordering where we denote the strict order by $<$. Then, the order topology is a topology on $X$ defined in the following equivalent ways.

In terms of subbasis

The order topology can be defined by means of the following subbasis:

• For $a\in X$, sets $(a,\mathrm{\infty })$, defined as sets of the form $\{x\mid a<x\}$
• For $a\in X$, sets $(-\mathrm{\infty },a)$, defined as sets of the form $\{x\mid x<a\}$

In terms of basis

The order topology can be defined by means of the following basis:

• For $a\in X$, sets $(a,\mathrm{\infty })$, defined as sets of the form $\{x\mid a<x\}$
• For $a\in X$, sets $(-\mathrm{\infty },a)$, defined as sets of the form $\{x\mid x<a\}$
• For $a,b\in X$ with $a<b$, sets $(a,b)$, defined as sets of the form $\{x\mid a<x<b\}$

Type of resultant topological space

We use the term linearly orderable space for a topological space that arises by taking the order topology on a linearly ordered set.