Continuum

continuum [kən′tin·yə·wəm]

(mathematics)

A compact, connected set.

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Continuum 

in mathematics a term used to denote structures that have certain properties of continuity [see (1) and (2) below for complete formulations] and to denote [not quite consistently in view of (2) below] the real numbers in discussing their cardinality.

(1) The most throughly studied continuous structure in mathematics is the system of real numbers, the number continuum. The properties of continuity of the system of real numbers can be characterized by different methods (using different “axioms of continuity”). If the concept of inequality (a < b) is used as the fundamental concept, then the continuity of the number continuum can be characterized, for example, by the following two statements: (a) between any two numbers a and b, where a < b, there lies at least one other number c (such that a < c < b ) and (b) if all the numbers are divided into two (nonempty) classes A and B such that every number a of A is less than any number b of B, then there exists either a greatest number in A or a least number in B (Dedekind’s axiom of continuity).

(2) In topology, which is the study of the geometry of continuity, the properties of the continuity of space or of any set are formulated using the concept of limit point. The fundamental concept of the connectivity of a set lying in a topological space (or of all space) is defined as follows: A set M is called connected if for any partition of M into two disjoint and nonempty subsets A and B there is at least one point that belongs to one of them and is a limit point of the other. In topology any connected compact Hausdorff space is called a continuum. Among sets on a line or in n -dimensional Euclidean space, the compact sets are the closed bounded sets. Thus, in Euclidean space a continuum can be defined as a connected closed bounded set. In particular, the only compact sets on the number line are closed intervals (that is, sets of numbers satisfying inequalities of the type a ≤ x ≤ b. In the strict sense of the topological definition of a continuum, the set of all real numbers is not a continuum.

(3) The cardinal number of the set of real numbers is called the cardinal number of the continuum and is denoted by either the Gothic letter C or by the Hebrew letter aleph א (without a subscript, in contrast to the other cardinal numbers). Every topological continuum has the same cardinal number C. The cardinal number C is greater than the cardinal number א₀ of countable sets. The continuum problem consists in deciding whether the cardinal number of the continuum is the next cardinal number after א₀