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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.
The following are all equivalent properties of a set S of natural numbers: . Semidecidability: The set S is computably enumerable. That is, S is the domain (co-range) of a partial computable function.
Every σ-compact space (the union of countably many compact spaces) is a Lindelöf space (every open cover has a countable subcover). [8] A metric space is σ-compact if and only if it is Lindelöf. [9] Every second-countable space (it has a countable base of open sets) is a separable space (it has a countable dense subset). [8]
Being countable implies being subcountable. In the appropriate context with Markov's principle , the converse is equivalent to the law of excluded middle , i.e. that for all proposition ϕ {\displaystyle \phi } holds ϕ ∨ ¬ ϕ {\displaystyle \phi \lor \neg \phi } .
If the axiom of choice holds, then a set is infinite if and only if it includes a countable infinite subset. If a set of sets is infinite or contains an infinite element, then its union is infinite. The power set of an infinite set is infinite. [3] Any superset of an infinite set is infinite. If an infinite set is partitioned into finitely many ...
An integer sequence is computable if there exists an algorithm that, given n, calculates a n, for all n > 0. The set of computable integer sequences is countable.The set of all integer sequences is uncountable (with cardinality equal to that of the continuum), and so not all integer sequences are computable.
The Cantor set is an example of an uncountable null set. It is uncountable because it contains all real numbers between 0 and 1 whose ternary form decimal expansion can be written using only 0’s and 2’s, and it is null because it is constructed by beginning with the closed interval of real numbers from 0 to 1 and multiplying the length by 2 ...
P satisfies the countable chain condition if every antichain in P is at most countable. This implies that V and V [ G ] have the same cardinals (and the same cofinalities). A subset D of P is called dense if for every p ∈ P there is some q ∈ D with q ≤ p .