- Continuum (mathematics)
In

mathematics , the word**"continuum**" has at least two distinct meanings, outlined in the sections below. For other uses seeContinuum .**Ordered set**The term

**"the continuum**" sometimes denotes thereal line . Somewhat more generally a continuum is alinearly 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 aleast 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)

*theaffinely extended real number system and order-isomorphic sets, for example theunit 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 thecardinality of the real line. Thecontinuum hypothesis is sometimes stated by saying that no cardinality lies between that of the continuum and that of thenatural numbers .See also

Suslin's problem .**Topology**In

point-set topology , a**continuum**is any nonempty compact connectedmetric space (or less frequently, acompact connectedHausdorff 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).**External links*** [

*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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