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Continuum (Mathematics)In mathematics, the word 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.
In point set topology, a continuum is a compact connected Hausdorff space. The interesting properties studied in the theory of continua arise from the fact that there exist nontrivial indecomposable continua (continua which cannot be written as the union of two proper subcontinua).
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