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Continuous set

From Encyclopedia of Mathematics
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A (totally) ordered set all proper cuts of which are Dedekind cuts, i.e. in any partition of into two non-empty subsets and such that every element of precedes every element of , either has a greatest element but no smallest element, or has no greatest element but 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