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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 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 ...
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.
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 ...
The decidable membership of = makes the set also not countable, i.e. uncountable. Beyond these observations, also note that for any non-zero number a {\displaystyle a} , the functions i ↦ f ( i ) ( i ) + a {\displaystyle i\mapsto f(i)(i)+a} in I → N {\displaystyle I\to {\mathbb {N} }} involving the surjection f {\displaystyle f} cannot be ...
Since this model is countable, its set of real numbers is countable. This consequence is called Skolem's paradox , and Skolem explained why it does not contradict Cantor's uncountability theorem: although there is a one-to-one correspondence between this set and the set of positive integers, no such one-to-one correspondence is a member of the ...
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 ...
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