Order topology

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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.