# Order topology

## Contents

## Definition

Suppose 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 defined in the following equivalent ways.

### In terms of subbasis

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

- For , sets , defined as sets of the form
- For , sets , defined as sets of the form

### In terms of basis

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

- For , sets , defined as sets of the form
- For , sets , defined as sets of the form
- For with , sets , defined as sets of the form

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.