Lecture
The word Continuum comes from the Latin continŭus (continuous).
The Continuum - is an uncountable set, or a set equinumerous with the set of real numbers in the interval (0,1), as well as the cardinal number of such a set (denoted 
or 
).
The set of real numbers in the interval (0,1), as in any other interval, is equinumerous with the set of all numbers on the number line, that is, one can construct a one-to-one mapping from the interval (0,1) onto the whole number line.
The set of points on the plane and in n-dimensional space, where n - is a finite number, also has the cardinality of the continuum.
From Cantor's theorem it follows that continuum sets are infinite. Therefore, if finite sets are added to them, their cardinality does not change as a result. Hence, the continuum sets include the sets 
- of irrational, 
- of complex and transcendental numbers.
The continuum is a more powerful set than the set of natural numbers. The question of the existence of a set with cardinality intermediate between the set of natural numbers and the set of the continuum is one of the unsolved problems of mathematics (see the Continuum hypothesis).
The continuum in set theory — is the cardinality (or cardinal number) of the set of all real numbers. It is denoted by the lowercase Latin letter c in fraktur script: . A set having the cardinality of the continuum is called a continuum set.
The term «continuum» may also denote the set of real numbers itself, or even any continuum set.
A function, variable or system is continuous, rather than discrete, if between any two points there is an infinite number of points and if, in addition, they have the completeness property; that is, if the distance between two points is measured by d, then for every number from 0 to d we can find a point whose distance to the first measures exactly this number. This applies, for example, to the real numbers, and also to space-time according to the theory of relativity.
According to Raymond Wilder (1965), there are four axioms that turn a set C and a relation < into a linear continuum:
These axioms characterize the order type of the real number line.
. Every continuum set contains a countable subset.. Every continuum set contains a subset of cardinality . The assumption that 
is called the continuum hypothesis.Examples of sets having the cardinality of the continuum:
.
(or 
), for example — the set of all complex numbers.
. (or ). (or ). (or ).The famous continuum hypothesis asserts that is also the second aleph number, . In other words, the continuum hypothesis asserts that there is no set 
whose cardinality lies strictly between and
It is now known that this statement is independent of the axioms of Zermelo–Fraenkel set theory with the axiom of choice (ZFC). That is, both the hypothesis and its negation are consistent with these axioms. In fact, for any nonzero natural number n the equality equals 
is independent of ZFC (the case 
is the continuum hypothesis). The same is true for most other alephs, although in some cases equality can be ruled out by König's theorem on the basis of cofinality (for example, 
). In particular, can be either or 
, where 
- is the first uncountable ordinal, so it can be either a successor cardinal or a limit cardinal, and either a regular cardinal or a singular cardinal.
In topology, a continuum is a connected and compact topological space. Continua arose as an attempt to characterize continuous functions as those that turn continua into continua. The idea did not catch on, but the term continued to be used, since in many areas of mathematics compact and connected sets are used. Some authors also require the Hausdorff property to hold.
From a topological point of view, in physics we speak of continuity, meaning a connected subset of Euclidean space.
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