Related:
Σ-compact space,
Axiom of countability,
Base (topology),
Category (mathematics),
Countable,
Dense (topology),
First-countable space,
Lattice (order),
Limit of a sequence,
Lindelöf space,
Mathematical object,
Mathematics,
Measure (mathematics),
Metric space,
Neighbourhood system,
Open cover,
Point (geometry),
Property,
Second-countable space,
Separable space,
Sequence,
Sequential space,
Set,
Sigma-finite,
Topological space,
In mathematics, an axiom of countability is a property of certain mathematical objects (usually in a category) that requires the existence of a countable set with certain properties, while without it such sets might not exist.
In mathematics, an axiom of countability is a property of certain mathematical objects (usually in a category) that requires the existence of a countable set with certain properties, while without it such sets might not exist.
In mathematics, a base (or basis) 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. Bases are useful because many properties of topologies can be reduced to statements about a base generating that topology, and because many topologies are most easily defined in terms of a base which generates them.
In mathematics, a category is an algebraic structure consisting of a collection of "objects", linked together by a collection of "arrows" that have two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. Objects and arrows may be abstract entities of any kind. Categories generalize monoids, groupoids and preorders. In addition, the notion of category provides a fundamental and abstract way to describe mathematical entities and their relationships. This is the central idea of category theory, a branch of mathematics which seeks to generalize all of mathematics in terms of objects and arrows, independent of what the objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. For more extensive motivational background and historical notes, see category theory and the list of category theory topics.In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor. The elements of a countable set can be counted one at a time — although the counting may never finish, every element of the set will eventually be associated with a natural number.
In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if any point in X can be "well-approximated" by points in A in the sense that any point in X is either an element or a limit point of A.