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Such sets are now called uncountable sets, and the size of infinite sets is treated by the theory of cardinal numbers, which Cantor began. Georg Cantor published this proof in 1891, [1] [2]: 20– [3] but it was not his first proof of the uncountability of the real numbers, which appeared in 1874.
One of the earliest results in set theory, published by Cantor in 1874, was the existence of different sizes, or cardinalities, of infinite sets. [2] An infinite set is called countable if there is a function that gives a one-to-one correspondence between and the natural numbers, and is uncountable if there is no such correspondence function.
The best known example of an uncountable set is the set of all real numbers; Cantor's diagonal argument shows that this set is uncountable. The diagonalization proof technique can also be used to show that several other sets are uncountable, such as the set of all infinite sequences of natural numbers (see: (sequence A102288 in the OEIS)), and the set of all subsets of the set ...
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. [a] Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time ...
The concept of countability led to countable operations and objects that are used in various areas of mathematics. For example, in 1878, Cantor introduced countable unions of sets. [67] In the 1890s, Émile Borel used countable unions in his theory of measure, and René Baire used countable ordinals to define his classes of functions. [68]
Every set X with the cocountable topology is Lindelöf, since every nonempty open set omits only countably many points of X. It is also T 1, as all singletons are closed. If X is an uncountable set then any two nonempty open sets intersect, hence the space is not Hausdorff. However, in the cocountable topology all convergent sequences are ...
It is called countable if the set of function and relation symbols in it is countable, and in general the cardinality of a signature is the cardinality of the set of all the symbols it contains. A first-order theory consists of a fixed signature and a fixed set of sentences (formulas with no free variables) in that signature.
The Cantor–Bendixson theorem states that closed sets of a Polish space X have the perfect set property in a particularly strong form: any closed subset of X can be written uniquely as the disjoint union of a perfect set and a countable set. In particular, every uncountable Polish space has the perfect set property, and can be written as the ...