Axiom of countability
Encyclopedia
In mathematics
, an axiom of countability is a property
of certain mathematical object
s (usually in a category
) that requires the existence of a countable set with certain properties, while without it such sets might not exist.
Important countability axioms for topological space
s:
Relations:
Other examples:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an axiom of countability is a property
Property
Property is any physical or intangible entity that is owned by a person or jointly by a group of people or a legal entity like a corporation...
of certain mathematical object
Mathematical object
In mathematics and the philosophy of mathematics, a mathematical object is an abstract object arising in mathematics.Commonly encountered mathematical objects include numbers, permutations, partitions, matrices, sets, functions, and relations...
s (usually in a category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
) that requires the existence of a countable set with certain properties, while without it such sets might not exist.
Important countability axioms for topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
s:
- sequential spaceSequential spaceIn topology and related fields of mathematics, a sequential space is a topological space that satisfies a very weak axiom of countability. Sequential spaces are the most general class of spaces for which sequences suffice to determine the topology....
: a set is open if every sequenceSequenceIn mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
convergentLimit of a sequenceThe limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
to a pointPoint (geometry)In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...
in the set is eventually in the set - first-countable spaceFirst-countable spaceIn 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...
: every point has a countable neighbourhood basis (local base) - second-countable spaceSecond-countable spaceIn topology, a second-countable space, also called a completely separable space, is a topological space satisfying the second axiom of countability. A space is said to be second-countable if its topology has a countable base...
: the topology has a countable baseBase (topology)In mathematics, a base B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T... - separable space: there exists a countable dense subspace
- Lindelöf spaceLindelöf spaceIn 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 subcover....
: every open cover has a countable subcover - σ-compact space: there exists a countable cover by compact spaces
Relations:
- Every first countable space is sequential.
- Every second-countable space is first-countable, separable, and Lindelöf.
- Every σ-compact space is Lindelöf.
- A metric spaceMetric spaceIn mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
is first-countable. - For metric spaces second-countability, separability, and the Lindelöf property are all equivalent.
Other examples:
- sigma-finite measure spaceMeasure (mathematics)In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
s - latticeLattice (order)In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
s of countable type