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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 ...
For each countable ordinal β choose a real number x β that is not in any of the sets S α for α < β, which is possible as the union of these sets has measure 0 so is not the whole of R. Then the uncountable set X of all these real numbers x β has only a countable number of elements in each set S α, so is a Sierpiński set.
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 ...
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 ...
In ZF, a set is infinite if and only if the power set of its power set is a Dedekind-infinite set, having a proper subset equinumerous to itself. [4] If the axiom of choice is also true, then infinite sets are precisely the Dedekind-infinite sets. If an infinite set is a well-orderable set, then it has many well-orderings which are non-isomorphic.
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 ...
Another example is the set of proper and bounded open intervals of real numbers with rational endpoints. ZF+AC ω suffices to prove that the union of countably many countable sets is countable. These statements are not equivalent: Cohen's First Model supplies an example where countable unions of countable sets are countable, but where AC ω ...
The counting measure can be defined on any measurable space (that is, any set along with a sigma-algebra) but is mostly used on countable sets. [ 1 ] In formal notation, we can turn any set X {\displaystyle X} into a measurable space by taking the power set of X {\displaystyle X} as the sigma-algebra Σ ; {\displaystyle \Sigma ;} that is, all ...
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