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...

closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel. For a topological...

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...

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...

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...

(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...

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...

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...

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...

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...

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...

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...

be extended to an arbitrary countable set A (e.g. the set of n-tuples of integers, the set of rational numbers, the set of formulas in some formal language...

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

theorems is his "revolutionary discovery" that the set of all real numbers is uncountably, rather than countably, infinite. This theorem is proved using Cantor's...

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...

known as the countable complement topology, is a topology that can be defined on any infinite set X {\displaystyle X} . In this topology, a set is open if...

translations would use these terms. Similarly, the terms for countable and uncountable sets come from countable and uncountable nouns.[citation needed] A crude sense...

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...

|\mathbb {N} |=\aleph _{0}} are called countable sets; these are either finite sets or countably infinite sets (sets of cardinality ℵ 0 {\displaystyle \aleph...

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...

topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly...

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...

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...

_{1}} are countable (finite or denumerable). Assuming the axiom of choice, the union of a countable set of countable sets is itself countable. So ℵ 1 {\displaystyle...

many sets and generalized series. In particular, this is why the definition of "countably additive" is rarely extended from countably many sets F 1 ,...

states that any Polish space can be written as the union of a countable set and a perfect set. Because any Gδ subset of a Polish space is again a Polish...

Kronecker objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included...

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