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. Equivalently, a set is countable...

Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union...

countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets...

null set is a Lebesgue measurable set of real numbers that has measure zero. This can be characterized as a set that can be covered by a countable union...

In mathematics, an Fσ set (said F-sigma set) is a countable union of closed sets. The notation originated in French with F for fermé (French: closed) and...

In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. The set of natural numbers (whose existence...

In set theory, a set is called hereditarily countable if it is a countable set of hereditarily countable sets. The inductive definition above is well-founded...

(over ZF) conditions: it has a countably infinite subset; there exists an injective map from a countably infinite set to A; there is a function f : A...

cocountable topology or countable complement topology on any set X consists of the empty set and all cocountable subsets of X, that is all sets whose complement...

uncountable ordinal is the set of all countable ordinals, expressed as ω1 or ⁠ Ω {\displaystyle \Omega } ⁠. In a well-ordered set, every non-empty subset...

mathematics, an axiom of countability is a property of certain mathematical objects that asserts the existence of a countable set with certain properties...

cocountable subset of a set X is a subset Y whose complement in X is a countable set. In other words, Y contains all but countably many elements of X. Since...

{\displaystyle \mathbb {N} } are called countable sets; these are either finite sets or countably infinite sets (sets of the same cardinality as ⁠ N {\displaystyle...

Moreover, every Borel set is Lebesgue-measurable. However, there are Lebesgue-measurable sets which are not Borel sets. Any countable set of real numbers has...

Neumann universe. So here it is a countable set. In 1937, Wilhelm Ackermann introduced an encoding of hereditarily finite sets as natural numbers. It is defined...

sets is a countable set. However, ZF with the ultrafilter lemma is too weak to prove that a countable union of countable sets is a countable set. The Hahn–Banach...

V=L Axiom of countability Every set is hereditarily countable Axiom of countable choice The product of a countable number of non-empty sets is non-empty...

cardinality as the set of the natural numbers, or | X | = | N | = ℵ 0 {\displaystyle \aleph _{0}} , is said to be a countably infinite set. Any set X with cardinality...

sets with E 1 ⊆ E 2 {\displaystyle E_{1}\subseteq E_{2}} then μ ( E 1 ) ≤ μ ( E 2 ) . {\displaystyle \mu (E_{1})\leq \mu (E_{2}).} For any countable sequence...

mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related...

mathematical field of descriptive set theory, a subset of a Polish space has the perfect set property if it is either countable or has a nonempty perfect subset...

In set theory, the complement of a set A, often denoted by A c {\displaystyle A^{c}} (or A′), is the set of elements not in A. When all elements in the...

algebra of sets, completed to include countably infinite operations. Axiomatic set theory Image (mathematics) § Properties Field of sets List of set identities...

related to set theory. Algebra of sets Axiom of choice Axiom of countable choice Axiom of dependent choice Zorn's lemma Axiom of power set Boolean-valued...

finite set is finite. All finite sets are countable, but not all countable sets are finite. (Some authors, however, use "countable" to mean "countably infinite"...

x}p(\omega ).} The points where the cdf jumps always form a countable set; this may be any countable set and thus may even be dense in the real numbers. A discrete...

German "Summe") on a set X is a nonempty collection Σ of subsets of X closed under complement, countable unions, and countable intersections. The ordered...

Every countable set is a strong measure zero set, and so is every union of countably many strong measure zero sets. Every strong measure zero set has Lebesgue...

infinite number of sets. A σ-additive set function is a function that has the additivity property even for countably infinite many sets, that is, μ ( ⋃ n...

Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are...