Difference between revisions of "Convex subset"
From Encyclopedia of Mathematics
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''of a partially ordered set'' | ''of a partially ordered set'' | ||
| − | A subset containing with any two elements | + | A subset containing with any two elements $a$ and $b$ the entire interval $[a,b]$ (cf. [[Interval and segment|Interval and segment]]). |
====Comments==== | ====Comments==== | ||
| − | A definition not involving the notion of interval is: A subset | + | 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|Order topology]]), then its connected subsets are convex. | 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|Order topology]]), then its connected subsets are convex. | ||
Revision as of 15:54, 9 April 2014
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://encyclopediaofmath.org/index.php?title=Convex_subset&oldid=12474
Convex subset. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Convex_subset&oldid=12474
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article