Difference between revisions of "Continuity axiom"
From Encyclopedia of Mathematics
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| − | The axiom expressing in one way or another the continuity of the set of real numbers (cf. [[ | + | The axiom expressing in one way or another the continuity of the set of real numbers (cf. [[Real number]]). The continuity axiom for the real numbers can be stated, e.g., in terms of cuts: Every cut of the real numbers is determined by some number ([[Dedekind theorem|Dedekind's axiom]]); in terms of nested closed intervals: Every family of nested closed intervals has a non-empty intersection ([[Cantor axiom|Cantor's axiom]]); in terms of upper or lower bounds of sets: Every non-empty set that is bounded from above has a finite least upper bound, and every set that is bounded from below has a greatest lower bound ([[Upper and lower bounds|Weierstrass' axiom]]); and in terms of the existence of limits of Cauchy sequences ([[Cauchy criteria#The elementary Cauchy criterion for sequences of real numbers|Cauchy criterion]]). |
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| + | See also [[Cut]]; [[Dedekind cut]]. | ||
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Latest revision as of 18:43, 15 April 2018
The axiom expressing in one way or another the continuity of the set of real numbers (cf. Real number). The continuity axiom for the real numbers can be stated, e.g., in terms of cuts: Every cut of the real numbers is determined by some number (Dedekind's axiom); in terms of nested closed intervals: Every family of nested closed intervals has a non-empty intersection (Cantor's axiom); in terms of upper or lower bounds of sets: Every non-empty set that is bounded from above has a finite least upper bound, and every set that is bounded from below has a greatest lower bound (Weierstrass' axiom); and in terms of the existence of limits of Cauchy sequences (Cauchy criterion).
Comments
See also Cut; Dedekind cut.
How to Cite This Entry:
Continuity axiom. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Continuity_axiom&oldid=43157
Continuity axiom. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Continuity_axiom&oldid=43157
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article