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Definition: Let X be a set with a simple order relation; assume X has more than one element. Let B be the collection of all sets of the following types: (1) All open intervals (a, b) in X. (2) All intervals of the form [a_0, b), where a_0 is the smallest element (if an) of X. (3) All intervals of the form (a, b_0], where b_0 is the largest element (if any) of X. The collection B is a basis for a topology on X, which is called the order topology.
Source: Topology (second edition) by James R. Munkres