Continuous set
From Encyclopedia of Mathematics
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
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