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