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Convex subset

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of a partially ordered set

A subset containing with any two elements $a$ and $b$ the entire interval $[a,b]$ (cf. Interval and segment).


Comments

A definition not involving the notion of interval is: A subset $A$ of a partially ordered set is convex if $a\leq b\leq c$ and $a,c\in A$ imply $b\in A$.

In the real line (with its usual ordering) the convex subsets are exactly the connected subsets (for the usual topology). This need not hold for more general ordered topological spaces. However, if a partially ordered set is equipped with the interval topology (cf. Order topology), then its connected subsets are convex.

How to Cite This Entry:
Convex subset. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Convex_subset&oldid=31462
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article