General topology

value that a function (or sequence) approaches as the argument (or index) approaches some value

right|thumb|A parabola, one of the simplest curves, after (straight) lines

set of points and set of neighborhoods that satisfy axioms relating those points to those neighborhoods

set that does not contain any of its boundary points

set in a topological space that contains an open superset of a given point or subset

topological space in which from every open cover of the space, a finite cover can be extracted

subset whose closure is the whole space

cluster point in a topological space

branch of topology dealing with general topological spaces

property limiting the "growth" of distances of outputs of a function uniformly across its domain

topological space that cannot be written as the disjoint union of two nonempty open subsets

set whose complement is an open set

dividing line between two areas or sets of points in a topological space; difference between the closure and the interior

point of a subset S where there exists a neighborhood around it that does not contain any other points of S

given a subset S of a topological space X, the biggest set of points in S not part of the boundary of S

family of subsets of a set whose union equals the whole set

topological space with a dense countable subset

collection of open sets that is sufficient for defining a topology

in a topological space, the smallest closed set containing a given set

topology on Cartesian products of topological spaces

theorem in topology and functional analysis

theorem about compact sets in Euclidean space

function whose range is a subset of the real numbers

inherited or induced topology

In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

topological space such that every open cover has a countable subcover

topological space whose every subset is both open and closed

in order theory, a nonempty, downward‐directed, upward‐closed subset of a preordered set

function that is discontinuous at rationals and continuous at irrationals

topology where the only open sets are the empty set and the entire space

subset that is both open and closed

coarse topology defined on algebraic varieties and schemes

topological space that is homeomorphic to a metric space

set whose closure has empty interior

preordered set whose every finite subset has an upper bound

topological generalization of the notion of a sequence

topological space in which every intersection of countably many dense open subsets is dense

in topology, a countable union of nowhere dense subsets

finite topological space with two points, only one of which is closed

a topology defined on the set of continuous maps between two topological spaces

given a space X, the space X ⊔ {∞}, topologized so that a set containing ∞ is open iff its complement is closed compact in X

comparison of topologies induced by the partial ordering on topologies on any given set

topological space consisting of equivalence classes of points in another topological space

separable, completely metrizable topological space

theorem

(for a point x) collection of all neighborhoods for the point x

a universal map from a topological space X to a compact Hausdorff space βX, such that any map from X to a compact Hausdorff space factors through βX uniquely; if X is Tychonoff, then X is a dense subspace of βX

In topology in mathematics, a subbase (or subbasis, prebase, prebasis) for the topology of a topological space is a subcollection B of \tau that generates \tau, in the sense that \tau is the smallest topology containing B as open sets. A slightly different definition is used by some authors, and there are other useful equivalent formulations of the definition; these are discussed below.

topology of topological vector spaces

topological space in which all connected components are singletons

topological space whose topology admits a countable base

topological space in which every point admits a countable neighborhood basis

property of topological spaces

mathematical term

property of certain mathematical objects (usually in a category) that asserts the existence of a countable set with certain properties. Without such an axiom, such a set might not probably exist.

certain topology on totally ordered sets

coarsest topology making certain functions continuous

category whose objects are topological spaces and whose morphisms are continuous maps

property which occurs on sufficiently small or arbitrarily small neighborhoods of points