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...

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...

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...

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

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...

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...

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...

  A set of formulas in the Lévy hierarchy ρ The rank of a set σ countable, as in σ-compact, σ-complete and so on Σ 1.  A sum of cardinals 2.  A set of...

infinite set. Any set can be given the cocountable topology, in which a set is defined as open if it is either empty or its complement is countable. When...

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...

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...

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

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...

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...

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"...

formulate probability theory on sets which are constrained to be measurable. The measurable sets on the line are iterated countable unions and intersections...

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...

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...

restricted set is countable and still forms a basis. Second-countability is a stronger notion than first-countability. A space is first-countable if each...

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...

of discrete random variables (random variables that take values on a countable set), while the PDF is used in the context of continuous random variables...

numbers are countable: As pointed out above, to show that a countable union of countable sets is itself countable requires the Axiom of countable choice....

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...

a countable set, while the set of real numbers and the set of complex numbers are both uncountable sets, and therefore larger than any countable set. All...

The empty set is not inhabited but generally deemed countable too, and note that the successor set of any countable set is countable. The set ω {\displaystyle...

Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any...

(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...

algebra"; also σ-field) on a set X is a nonempty collection Σ of subsets of X closed under complement, countable unions, and countable intersections. The ordered...