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:

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

Note that for the purpose of constructing a basis, we can ignore open subsets of types (1) and (2) and simply consider open subsets of type (3) because any open subset of type (1) or (2) is expressible as a union of open subsets of type (3). In other words, the subsets of type (3) alone form a basis for the same topology.

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.