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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the word continuum has at least two distinct meanings, outlined in the sections below. For other uses see Continuum
Continuum

Continuum can refer to:* Continuum , anything that goes through a gradual transition from one condition, to a different condition, without any abrupt changes or "discontinuities"....
.

term the continuum sometimes denotes the real line
Real line

In mathematics, the real line is simply the set R of singleton real numbers.However, this term is usually used when R is to be treated as a space of some sort, such as a topological space or a vector space....
. Somewhat more generally a continuum is a linearly 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 a least upper bound.

Examples in addition to the real numbers:

cardinality of the continuum
Cardinality of the continuum

In mathematics, the cardinality of the continuum, sometimes also called the power of the continuum, is the size of the Set of real numbers ....
 is the cardinality
Cardinality

In mathematics, the cardinality of a set is a measure of the "number of Element of the set". For example, the set A = contains 3 elements, and therefore A has a cardinality of 3....
 of the real line.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the word continuum has at least two distinct meanings, outlined in the sections below. For other uses see Continuum
Continuum

Continuum can refer to:* Continuum , anything that goes through a gradual transition from one condition, to a different condition, without any abrupt changes or "discontinuities"....
.

Ordered set

The term the continuum sometimes denotes the real line
Real line

In mathematics, the real line is simply the set R of singleton real numbers.However, this term is usually used when R is to be treated as a space of some sort, such as a topological space or a vector space....
. Somewhat more generally a continuum is a linearly 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 a least upper bound.

Examples in addition to the real numbers:
  • sets which are order-isomorphic
    Order isomorphism

    In the mathematics field of order theory an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets ....
     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)
  • the affinely extended real number system and order-isomorphic sets, for example the unit interval
    Unit interval

    In mathematics, the unit interval is the interval [0,1], that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1....
  • the set of real numbers with only +8 or only -8 added, and order-isomorphic sets, for example a half-open interval
  • the long line
    Long line (topology)

    In topology, the long line is a topological space analogous to the real line, but much longer. Because it behaves locally just like the real line, but has different large-scale properties, it serves as one of the basic counterexamples of topology....


Cardinality of the continuum

The cardinality of the continuum
Cardinality of the continuum

In mathematics, the cardinality of the continuum, sometimes also called the power of the continuum, is the size of the Set of real numbers ....
 is the cardinality
Cardinality

In mathematics, the cardinality of a set is a measure of the "number of Element of the set". For example, the set A = contains 3 elements, and therefore A has a cardinality of 3....
 of the real line. The continuum hypothesis
Continuum hypothesis

In mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor, about the possible sizes of infinite Set . Cantor introduced the concept of cardinal number to compare the sizes of infinite sets, and he gave two proofs that the cardinality of the set of integers is strictly smaller than that of the set of real numbers....
 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
Suslin's problem

In mathematics, Suslin's problem is a question about totally ordered sets posed by Mikhail Yakovlevich Suslin in the early 1920s.It has been shown to be independence of the standard axiomatic system of set theory known as ZFC: the statement can neither be proven nor disproven from those axioms....
.

Topology

In point-set topology, a continuum is any nonempty compact
Compact space

In mathematics, a topological space is called compact if each of its open covers has a finite set subcover.Note: Some authors such as Nicolas Bourbaki use the term "quasi-compact" for this instead, and reserve the term "compact" for topological spaces that are both Hausdorff spaces and "quasi-compact"....
 connected
Connected space

In topology and related branches of mathematics, a connected space is a topological space which cannot be represented as the disjoint union of two or more nonempty open subsets....
 metric space
Metric space

In mathematics, a metric space is a Set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space....
 (or less frequently, a compact connected
Connected space

In topology and related branches of mathematics, a connected space is a topological space which cannot be represented as the disjoint union of two or more nonempty open subsets....
 Hausdorff space
Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhood ....
).

A continuum that contains more than one point (and thus infinitely many by its connectedness and Hausdorff properties) is called nondegenerate
Degeneracy (mathematics)

In mathematics, a degenerate case is a limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class....
. 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
Indecomposable continuum

In point-set topology, an indecomposable continuum is a Continuum that is not the union of any two of its proper subcontinua. The pseudo-arc is an example of a Glossary of general topology indecomposable continuum....
 (continua which cannot be written as the union of two proper subcontinua).

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