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Difference between revisions of "Continuous set"

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A (totally) [[Ordered set|ordered set]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025760/c0257601.png" /> all proper cuts of which are Dedekind cuts, i.e. in any partition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025760/c0257602.png" /> into two non-empty subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025760/c0257603.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025760/c0257604.png" /> such that every element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025760/c0257605.png" /> precedes every element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025760/c0257606.png" />, either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025760/c0257607.png" /> has a greatest element but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025760/c0257608.png" /> no smallest element, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025760/c0257609.png" /> has no greatest element but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025760/c02576010.png" /> has a smallest element.
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A (totally) [[Ordered set|ordered set]] 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.
  
  

Revision as of 15:48, 18 June 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
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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