- Continuum (mathematics)
mathematics, the word "continuum" has at least two distinct meanings, outlined in the sections below. For other uses see Continuum.
The term "the continuum" sometimes denotes the
real line. Somewhat more generally a continuum is a linearly ordered setof more than one element that is "densely ordered", i.e., between any two members there is another, and it lacks gaps in the sense that every non-empty subset with an upper bound has a least upper bound.
Examples in addition to the real numbers:
*sets which are order-isomorphic to the set of real numbers, for example a real open interval, and the same with half-open gaps (note that these are not gaps in the above-mentioned sense)
affinely extended real number systemand order-isomorphic sets, for example the unit interval
*the set of real numbers with only +∞ or only -∞ added, and order-isomorphic sets, for example a half-open interval
*the long line
Cardinality of the continuum
cardinality of the continuum" is the cardinalityof the real line. The continuum hypothesisis sometimes stated by saying that no cardinality lies between that of the continuum and that of the natural numbers.
point-set topology, a continuum is any nonempty compact connected metric space(or less frequently, a compactconnected Hausdorff space).
A continuum that contains more than one point (and thus infinitely many by its connectedness and Hausdorff properties) is called nondegenerate. Continuum theory refers to the branch of topology related to the study of continua. One interesting subject in continuum theory is the existence of nontrivial "indecomposable continua" (continua which cannot be written as the union of two proper subcontinua).
* [http://web.mst.edu/~continua/ Open problems in continuum theory]
* [http://www.karlin.mff.cuni.cz/~pyrih/e/ Examples in continuum theory]
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