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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 ...
In mathematics, a Sierpiński set is an uncountable subset of a real vector space whose intersection with every measure-zero set is countable. The existence of Sierpiński sets is independent of the axioms of ZFC. Sierpiński showed that they exist if the continuum hypothesis is true. On the other hand, they do not exist if Martin's axiom for ...
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 ...
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 ...
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 ...
There are many equivalent forms to the axiom of countable choice, in the sense that any one of them can be proven in ZF assuming any other of them. They include the following: [8] [9] Every countable collection of non-empty sets has a choice function. [8] Every infinite collection of non-empty sets has an infinite sub-collection with a choice ...
A set is countable if it can be enumerated, that is, if there exists an enumeration of it. Otherwise, it is uncountable. For example, the set of the real numbers is uncountable. A set is finite if it can be enumerated by means of a proper initial segment {1, ..., n} of the natural numbers, in which case, its cardinality is n.
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