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For a countable (or finite) set, the argument of the proof given above can be illustrated by constructing a table in which each row is labelled by a unique from = {,, …}, in this order. is assumed to admit a linear order so that such table can be constructed.
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]
A generalized form of the diagonal argument was used by Cantor to prove Cantor's theorem: for every set S, the power set of S—that is, the set of all subsets of S (here written as P(S))—cannot be in bijection with S itself. This proof proceeds as follows: Let f be any function from S to P(S). It suffices to prove f cannot be surjective.
A still weaker example is the axiom of countable choice (AC ω or CC), which states that a choice function exists for any countable set of nonempty sets. These axioms are sufficient for many proofs in elementary mathematical analysis , and are consistent with some principles, such as the Lebesgue measurability of all sets of reals, that are ...
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 ...
In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets.It states: There is no set whose cardinality is strictly between that of the integers and the real numbers.
first-countable space: every point has a countable neighbourhood basis (local base) second-countable space: the topology has a countable base; separable space: there exists a countable dense subset; Lindelöf space: every open cover has a countable subcover; σ-compact space: there exists a countable cover by compact spaces