In mathematics, the word continuum has at least two distinct meanings, outlined in the sections below. For other uses see Continuum.
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Ordered set
The term the continuum sometimes denotes the real line. Somewhat more generally a continuum is a linearly ordered set of 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)
- the affinely extended real number system and 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
The cardinality of the continuum is the cardinality of the real line. The continuum hypothesis is sometimes stated by saying that no cardinality lies between that of the continuum and that of the natural numbers.
See also Suslin's problem.
Topology
- See also: Linear continuum
In point-set topology, a continuum is any nonempty compact connected metric space (or less frequently, a compact connected 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).
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