quality:correction, structural:summarization
Order topology

Convexity can be extended for a totally ordered set X endowed with the order topology.

Let Y ⊆ X. The subspace Y is a convex set if for each pair of points a, b in Y such that a ≤ b, the interval = { x ∈ X | a ≤ x ≤ b } is contained in Y. That is, Y is convex if and only if for all a, b in Y, a ≤ b implies ⊆ Y.

A convex set is not connected in general: a counter-example is given by the subspace {1,2,3} in Z, which is both convex and not connected.