Continuum (mathematics)
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In mathematics, the word continuum has at least two distinct meanings, outlined in the sections below. For other uses see Continuum.
Ordered set
The term continuum sometimes denotes the real line. Somewhat more generally a continuum is a linearly ordered set that is "densely ordered", i.e., between any two members there is another, and lacks gaps, i.e., every non-empty subset with an upper bound has a least upper bound. By that definition, the long line is a continuum, as are various other sets besides the real line.
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.
Compact and connected Hausdorff space
In point-set topology, a continuum is any nontrivial compact connected Hausdorff space (or sometimes more specifically a compact connected metric 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).Template:Topology-stub