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...

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...

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...

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

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...

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...

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...

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...

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...

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...

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"...

(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...

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...

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...

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...

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...

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...

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...

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...

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...

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....

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

set is a subset of a topological space that is a countable intersection of open sets. The notation originated from the German nouns Gebiet 'open set'...

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...

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...