J. Munkres. Topology (2013) Chapter 2: Topological Spaces and Continuous Functions. 14: The Order Topology

Definition

Let \(X\) be a linearly ordered set with more than one element. The order topology on \(X\) is the topology generated by the basis consisting of:

  1. All open intervals
  2. All intervals of the form \([\bot, b)\) where \(\bot\) is the smallest element of \(X\), if this exists.
  3. All intervals of the form \((a, \top]\) where \(\top\) is the greatest element of \(X\), if this exists.

Example 1

The standard topology on \(\mathbb{R}\) is the order topology.

The order topology is the topology generated by the subbasis of all open rays.

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