The term "continuum" has (at least) two distinct technical meanings in mathematics.

The first is a compact connected metric space (Kuratowski 1968; Lewis 1983, pp. 361-394; Nadler 1992; Prajs and Charatonik).

The second is the nondenumerable set of real numbers, denoted . The continuum satisfies

and

where is ℵ₀ (Aleph-0) and is a positive integer. It is also true that

for . However,

is a set larger than the continuum. Paradoxically, there are exactly as many points on a line (or line segment) as in a plane, a three-dimensional space, or finite hyperspace, since all these sets can be put into a one-to-one correspondence with each other.

The continuum hypothesis, first proposed by Georg Cantor, holds that the cardinal number of the continuum is the same as that of ℵ₁. The surprising truth is that this proposition is undecidable, since neither it nor its converse contradicts the tenets of set theory.

See also

Aleph-0, Aleph-1, Compact Space, Connected Space, Continuum Hypothesis, Decomposable Continuum, Denumerable Set, Hereditarily Decomposable Continuum, Indecomposable Continuum, Irrational Number, Metric Space, Rational Number

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References

Kuratowski, K. Topology, Vol. 2. New York: Academic Press, 1968.Lewis, W. "Continuum Theory Problems." Topology Proc. 8, 361-394, 1983.Nadler, S. B. Jr. Continuum Theory. New York: Dekker, 1992.Prajs, J. R. and Charatonik, W. J. (Eds.). "Open Problems in Continuum Theory." http://web.umr.edu/~continua/.

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Continuum

Cite this as:

Weisstein, Eric W. "Continuum." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Continuum.html

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