Editing Convex set (section)

=== Order topology ===
Convexity can be extended for a [[totally ordered set]] {{mvar|X}} endowed with the [[order topology]].<ref>[[James Munkres|Munkres, James]]; ''Topology'', Prentice Hall; 2nd edition (December 28, 1999). {{ISBN|0-13-181629-2}}.</ref>

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

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