Difference between revisions of "Continuous set"
From Encyclopedia of Mathematics
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| − | A | + | A [[totally ordered set]] $X$ all proper cuts of which are [[Dedekind cut]]s, i.e. in any partition of $X$ into two non-empty subsets $X_1$ and $X_2$ such that every element of $X_1$ precedes every element of $X_2$, either $X_1$ has a greatest element but $X_2$ no smallest element, or $X_1$ has no greatest element but $X_2$ has a smallest element. |
| + | ====Comments==== | ||
| + | Thus, a continuous set is a [[conditionally-complete lattice]] which is a dense total order. The phrase "continuous set" is not used in the Western literature. | ||
| − | + | [[Category:Order, lattices, ordered algebraic structures]] | |
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Latest revision as of 14:20, 18 October 2014
A totally ordered set $X$ all proper cuts of which are Dedekind cuts, i.e. in any partition of $X$ into two non-empty subsets $X_1$ and $X_2$ such that every element of $X_1$ precedes every element of $X_2$, either $X_1$ has a greatest element but $X_2$ no smallest element, or $X_1$ has no greatest element but $X_2$ has a smallest element.
Comments
Thus, a continuous set is a conditionally-complete lattice which is a dense total order. The phrase "continuous set" is not used in the Western literature.
How to Cite This Entry:
Continuous set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Continuous_set&oldid=32251
Continuous set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Continuous_set&oldid=32251
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article