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In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers ... the set of all continuous functions from ...
The cardinality of the continuum is the size of the set of real numbers. The continuum hypothesis is sometimes stated by saying that no cardinality lies between that of the continuum and that of the natural numbers , ℵ 0 {\displaystyle \aleph _{0}} , or alternatively, that c = ℵ 1 {\displaystyle {\mathfrak {c}}=\aleph _{1}} .
As is standard in set theory, we denote by the least infinite ordinal, which has cardinality ; it may be identified with the set of natural numbers.. A number of cardinal characteristics naturally arise as cardinal invariants for ideals which are closely connected with the structure of the reals, such as the ideal of Lebesgue null sets and the ideal of meagre sets.
Axiomatic set theory was at that point not yet formulated. Kurt Gödel proved in 1940 that the negation of the continuum hypothesis, i.e., the existence of a set with intermediate cardinality, could not be proved in standard set theory. [2]
There are two ways to define the "cardinality of a set": The cardinality of a set A is defined as its equivalence class under equinumerosity. A representative set is designated for each equivalence class. The most common choice is the initial ordinal in that class. This is usually taken as the definition of cardinal number in axiomatic set theory.
The cardinality of the set () ... A construction adding at most countably many points is ... The set of all real-valued continuous functions on a separable ...
If X is a separable, complete metric space with no isolated points, the cardinality of X is exactly . If X is a locally compact Hausdorff space with no isolated points, there is an injective function (not necessarily continuous) from Cantor space to X , and so X has cardinality at least 2 ℵ 0 {\displaystyle 2^{\aleph _{0}}} .
Cardinal functions are widely used in topology as a tool for describing various topological properties. [2] [3] Below are some examples.(Note: some authors, arguing that "there are no finite cardinal numbers in general topology", [4] prefer to define the cardinal functions listed below so that they never taken on finite cardinal numbers as values; this requires modifying some of the ...