mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, an axiom of countability is a property of certain

mathematical object A mathematical object is an abstract concept arising in mathematics. Typically, a mathematical object can be a value that can be assigned to a Glossary of mathematical symbols, symbol, and therefore can be involved in formulas. Commonly encounter ...

s that asserts the existence of a

countable set In 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'' if there exists an injective function from it into the natural numbe ...

with certain properties. Without such an axiom, such a set might not provably exist.

Important examples

Important countability axioms for

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

s include:. * sequential space: a set is closed if and only if every convergent

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...

in the set has its limit point in the set *

first-countable space In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base ...

: every point has a countable neighbourhood basis (local base) * second-countable space: the topology has a countable base *

separable space In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence ( x_n )_^ of elements of the space such that every nonempty open subset of the space contains at least one elemen ...

: there exists a countable dense subset *

Lindelöf space In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of ''compactness'', which requires the existence of a ''finite'' sub ...

: every

open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a family of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\su ...

has a countable subcover *

σ-compact space In mathematics, a topological space is said to be ''σ''-compact if it is the union of countably many compact subspaces. A space is said to be ''σ''-locally compact if it is both ''σ''-compact and (weakly) locally compact. That terminology ...

: there exists a countable cover by compact spaces

Relationships with each other

These axioms are related to each other in the following ways: *Every first-countable space is sequential. *Every second-countable space is first countable, separable, and Lindelöf. *Every σ-compact space is Lindelöf. *Every

metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...

is first countable. *For metric spaces, second-countability, separability, and the Lindelöf property are all equivalent.

Related concepts

Other examples of mathematical objects obeying axioms of countability include sigma-finite

measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...

s, and lattices of countable type.

References

{{sia, mathematics General topology Mathematical axioms