n
(topology, "of" a subset of a topological space) Given a subset S of a topological space X, a point x whose every neighborhood contains at least one point distinct from x that belongs to S.
n
(topology) A topology in which the intersection of any family of open sets is an open set.
adj
(topology) A regular Hausdorff space is said to be angelic if the closure of each relatively countably compact set A is compact and the closure consists of the limits of sequences in A.
adj
(topology) Having the property that for all points x distinct from a specified point p, there is a compact connected neighborhood of p that does not include x.
adj
(of a topological space) Such that every pair of points in the space comprises the boundary of some arc embedded in the space.
n
(differential geometry, topology) A family of coordinate charts that cover a manifold.
n
(mathematics) A full group fixing any ordered pair of components of the spread of a vector space over a prime field.
n
A topological space such that every intersection of a countable collection of open dense sets in the space is also dense.
n
(topology) The set of sets from which a topology is generated.
n
(topology) A collection of subsets ("basis elements") of a set, such that this collection covers the set, and for any two basis elements which both contain an element of the set, there is a third basis element contained in the intersection of the first two, which also contains that element.
adj
(topology) homeomorphic
n
(topology) A bitopological space.
n
(topology) The property held by some topological spaces that if a subset of such a space has an infinite quantity of points then the subset has at least one accumulation point.
n
(topology) (of a set) The set of points in the closure of a set S, not belonging to the interior of that set.
n
(algebraic topology) The image of the (k + 1)ᵗʰ boundary homomorphism (for a given chain complex).
n
(topology) A nontrivial link that becomes a set of trivial unlinked circles if any one component is removed. In other words, cutting any loop frees all the other loops.
n
(mathematics) Topological space composed of a base space and fibers projected to the base space.
n
(topology) A locally connected continuum that is the closure of the union of an at most countable number of spheres and simple arcs, of which every simple closed contour is contained in one and only one of the spheres.
n
(mathematical analysis, topology) A subset of an interval formed by recursively removing an interval in the middle of every connected component of the set.
n
(topology) A method of adjoining a chain of degree p with a cochain of degree q, such that q ≤ p, to form a composite chain of degree p − q.
n
(topology) A space for which notions of a Cauchy sequence and completeness exist.
n
(mathematics) A topological space that is the quotient of a free action (of the specified group) on a weakly contractible space.
adj
(topology, of a set in a topological space) Both open and closed.
adj
(topology, of a set) Having an open complement.
n
(topology) A map from the circle, S¹, to a topological space.^((McLarty, p. 4))
n
(mathematics) Alternative form of closed curve [(topology) A map from the circle, S¹, to a topological space.^((McLarty, p. 4))]
n
(topology) A set whose complement is open.
n
(topology, of a set) The smallest closed set which contains the given set.
n
(topology) A topology consisting of the empty set and all cofinite subsets of the underlying set.
n
(topology, mathematical analysis) A set of sets.
adj
(topology, not comparable, of a set) Such that every open cover of the given set has a finite subcover.
n
(mathematics) Any topological subset of Euclidean space that is a compact set
n
(topology) The space resulting from any such procedure.
n
(mathematics) A compact topological space.
n
(topology) A connected component of the complement in a manifold of a lamination of that manifold.
adj
(topology, of a surface embedded in a 3-dimensional manifold) Containing a circle that does not serve as the boundary of a disk that lies in the surface, but does serve as boundary of a disk that lies in the ambient manifold.
n
(topology) (said to be "of" a given subset of a topological space) a point whose every neighborhood contains uncountably many points belonging to the given subset
n
(topology) A space formed by taking the direct product of a given space with a closed interval and identifying all of one end to a point.
adj
(mathematics) Given a binary operation →_𝛽 on a set A, and its reflexive, transitive closure ↠_𝛽 , then, for all a1, a2, and a3 in A, if a1 →_𝛽 a2 and a1 →_𝛽 a3, then there must exist an a4 in A such that a2 ↠_𝛽 a4 and a3 ↠_𝛽 a4.
adj
(mathematics, topology, of a topological space) That cannot be partitioned into two nonempty open sets.
n
(topology and graph theory) A connected subset that is, moreover, maximal with respect to being connected.
n
(mathematics) Any topological space which cannot be written as the disjoint union of two or more nonempty open spaces.
n
(mathematics) A connectopic mapping
n
(topology) The branch of topology devoted to the study of continua, a type of metric space.
n
(topology) A set (more often known as a family) of sets, whose union contains the given set.
n
(mathematics) A map from a topological space onto another by local homeomorphisms of disjoint preimages.
n
(topology) A point of a connected space whose removal causes the resulting space to be disconnected.
n
(topology, algebraic topology) A chain whose boundary is zero.
n
(topology) A subspace such that there is a retraction onto it of the ambient space homotopic to the identity function on the ambient space.
adj
(mathematics, topology) Being a subset of a topological space that approximates the space well. See the Wikipedia article on dense sets for a mathematical definition.
adj
(mathematics, of a subset of a topological space) Having no isolated points.
n
(topology) A particular extension of the idea of gluing.
n
(topology) A structure defined for a (topological) manifold so that it supports differentiation of functions defined on it.
n
(topology) A topological space analogously formed from two or more (up to an infinite number of) topological spaces.
adj
(mathematics, of a topological space) That can be partitioned into two nonempty subsets which are both open and closed.
adj
(topology) Having each singleton subset open: said of a topological space or a topology.
n
(topology) A set of points of a topological space such that each point in the set is an isolated point, i.e. a point that has a neighborhood that contains no other points of the set.
n
(topology) A topology on a set consisting of all subsets of that set.
n
(topology) The disjoint union of the underlying sets of a given family of topological spaces, equipped with a topology.
n
(topology) A topology which is applicable to the disjoint union of a given set of topological spaces and is the largest (most inclusive) topology that preserves the continuity of each contributing space as represented in the union.
n
(topology) A dichotomous topology; A topology that includes two structures: one describing the properties related to openness of sets, the other describing the properties related to closedness of sets.
n
(mathematics) Of a vector in an inner product space, the linear functional corresponding to taking the inner product with that vector. The set of all duals is a vector space called the dual space.
n
(topology) The topological space obtained by identifying the sides comprising the boundary of a triangle, two sides with consistent orientation and the third with the reverse orientation.
n
(mathematics, number theory) A rigid analytic curve that parametrizes certain p-adic families of modular forms.
n
(mathematics) A topological space that has precisely one nontrivial homotopy group.
n
(mathematics) A Riemann space in which the contracted curvature tensor is proportional to the metric tensor
n
(topology) A binary relation in a uniform space which generalises the notion of two points being no farther apart than a given fixed distance; a uniform neighbourhood.
n
(topology) Given a collection S of subsets of a set X: a subcollection S^* of S such that each element in X is contained in exactly one subset in S^*.
n
(topology) The fact that, under certain hypotheses, the homology of a space relative to a subspace is unchanged by the identification of a subspace of the latter to a point.
n
(topology, idiomatic) A space obtained from another by identification of points that are equivalent to one another in some equivalence relation.
n
(American spelling, topology, category theory) A topological space E with a base space B and a fiber space F such that any point x ∈ B has a neighborhood N that is homeomorphic to the product space B × F (that is, the space is locally the product space B × F, although its global structure can be quite different).
n
(topology) A simplicial complex where every collection of pairwise adjacent vertices spans a simplex.
n
(mathematics) Any Riemann space in which the metric tensors are constants throughout
n
(mathematics) Any discrete group of isometries of the hyperbolic plane.
n
(topology, algebraic geometry) A property that is true in some dense open subset of a given set.
n
(topology) A topological space which represents some graph (ordered pair of sets) and which is constructed by representing the vertices as points and the edges as copies of the real interval [0,1] (where, for any given edge, 0 and 1 are identified with the points representing the two vertices) and equipping the result with a particular topology called the graph topology.
n
(topology) A topological space X (generally assumed to be connected) together with a continuous map μ : X × X → X with an identity element e such that μ(e, x) = μ(x, e) = x for all x in X.
n
(topology) A topological space homeomorphic to a ball but viewed as a product of two lower-dimensional balls.
n
(topology) A top-dimensional submanifold of Euclidean space comprising a ball with handles attached to it along its boundary.
adj
(of a topological space) Such that any two distinct points have disjoint neighborhoods.
n
(topology) A topological space in which for any two distinct points x and y, there is a pair of disjoint open sets U and V such that x∈U and y∈V.
n
(topology) A metrizable topological space which is the product of a countably infinite number of closed intervals.
n
(functional analysis) A generalized Euclidean space in which mathematical functions take the place of points; crucial to the understanding of quantum mechanics and other applications.
n
A manifold or topological space X on which G acts by symmetry in a transitive way.
n
(topology) A manifold whose homology is the same as that of some sphere of the same dimension
n
(topology) A continuous deformation of one continuous function or map to another.
n
(mathematics) A knot in a three-dimensional continuous unit vector field that cannot be unknotted without cutting
adj
(mathematics, of a graph) Having an edge set that spans a matroid defined on its edges via exterior algebra.
n
The topology of hyperspace.
n
(topology) A space obtained from another by identification of points that are equivalent to one another in some equivalence relation.
n
(topology) Given a set X, a family of topological spaces (M_i,T_i)_(i∈I) and a family of functions f_i:X→M_i, the initial topology is the coarsest topology on X such that all f_i are continuous.
adj
(topology, of a manifold) Not containing a sphere of codimension 1 that is not the boundary of a ball.
n
The vector product of the translational and angular displacements of each joint of a robotic link; all the places that a robotic arm can access.
n
(geometry) The conjecture that, in any tiling of n-dimensional Euclidean space by identical hypercubes, there are two hypercubes that share an entire (n − 1)-dimensional face with each other.
adj
(of a topological space) Such that any two distinct points are topologically distinguishable, i.e., there is an open set containing one of the points which does not contain the other point.
n
(topology) A topological space for which any pair of distinct points are also topologically distinct, i.e., there is an open set which contains one of the points but not the other one.
n
(topology) A foliation of a closed subset of a manifold by subspaces of one dimension less.
n
(mathematics) A topological space that is the quotient of a laminated or foliated manifold by identification of each leaf, and each closed complementary region, to a point.
n
(topology) Given a subset S of a given topological space T, any point p whose every neighborhood contains some point, distinct from p, which belongs to S.
adj
(topology, of a topological space) Such that any open cover has a countable subcover.
n
(topology) A topological space such that any open cover of the space has a countable subcover.
adj
(topology) Of a topological space: such that, for every point of the space, there is a neighborhood of that point whose closure is compact.
n
(mathematics) A topological space that looks locally like the "ordinary" Euclidean space ℝⁿ and is Hausdorff.
n
(topology) A set (a "universe") that consists of selected mathematical objects that are treated as points, and selected relationships between them.
n
(topology) A topological game invented by the Polish mathematician Mazur in order to illustrate the difference between sets of the first category and second category.
adj
(topology) Of a topological space: such that every open cover has a point finite open refinement. That is, given any open cover of the topological space, there is a refinement which is again an open cover with the property that every point is contained only in finitely many sets of the refining cover.
adj
(topology) Of a topological space: for which a metric exists that will induce the original topology.
n
(mathematics) A kind of generalization of a vector bundle.
adj
(of a compact topological space with boundary) Having had some but not all of its boundary removed.
n
(mathematics) n-dimensional Euclidean space
n
(computing, graph theory) A set of nodes.
n
(topology) A regular space with the additional property that for every disjoint pair of closed sets in that space, there is a disjoint pair of open sets which contain the closed sets, respectively.
adj
(mathematics, topology, of a set) Which is part of a predefined collection of subsets of X, that defines a topological space on X.
n
(mathematics, topology) An open book decomposition.
n
(topology) A cover whose members are all open sets.
n
(topology) Most generally, a member of the topology of a given topological space.
n
(topology) A topological space in which every small enough neighborhood is homeomorphic to a quotient of real space by the action of a finite group.
n
(topology) A set that is the union of subsets (called segments), each of which is totally ordered, such that the segments fit together a certain way.
n
(mathematics) A type of vector field defined on an open set of such that every vector in the field is tangent to the corresponding point in the set.
adj
(topology) Pertaining to the category of all metric spaces and all CW-complexes.
n
(topology) A continuous map f from the unit interval I=[0,1] to a topological space X.
n
(less formally) the underlying set of such a topological space.
adj
(topology) Having a defining atlas such that there is a piecewise linear homeomorphism between any pair of intersecting atlas charts: said of manifolds
n
(topology) Piecewise linear.
n
(mathematics) Any of a group of simple quadratic relations that are satisfied by the Plücker embedding.
n
The general field of topology, not restricting attention to specific classes of spaces, and not using algebraic topology.
adj
(topology, algebraic topology, of a topological space) That has a named, but otherwise arbitrary, point (called the basepoint) that remains unchanged during subsequent discussion and is kept track of during all operations.
n
(topology) A separable, completely metrizable topological space
adj
(topology) Totally bounded; Having a cover that consists of finitely many finite subsets.
adj
(topology, of a space) In which any two topologically distinguishable points can be separated by neighbourhoods
n
(mathematics) A filter subset or a preclosure operator which acts as a distinct prespace (pre-topological space).
n
(topology) The topology of the Cartesian product of two or more topological spaces which is generated from a basis whose elements are Cartesian products of open subsets of the original spaces, such that for each such Cartesian product only a finite quantity of such subsets are proper.
adj
(topology) Pertaining to the generalization of Anosov diffeomorphism of the torus introduced by William Thurston.
n
A special type of topological space that looks like a manifold at most of its points, but may contain singularities.
adj
(topology, of a space) In which has every open cover has a finite subcover
adj
Alternative form of quasi-compact [(topology, of a space) In which has every open cover has a finite subcover]
n
(topology and algebra) A space obtained from another by identification of points that are equivalent to one another in some equivalence relation.
n
(topology) A Hausdorff space with the additional property that for every closed set of that space and every point disjoint from that set, there are a disjoint pair of open sets which contain the closed set and the point, respectively.
n
(topology) subspace topology
n
(computer graphics) Modification of the topology of an object.
n
(mathematics) A continuous function from a topological space onto a subspace which is the identity on that subspace.
adj
Alternative form of second-countable [(topology) Such that its topology has a countable base, said of a topological space.]
adj
(topology) Such that its topology has a countable base, said of a topological space.
n
(topology, chiefly US) A fiber bundle whose fiber is the circle and whose base space is a two-dimensional orbifold
n
(topology, US) Alternative spelling of Seifert fiber space [(topology, chiefly US) A fiber bundle whose fiber is the circle and whose base space is a two-dimensional orbifold]
n
(topology, Britain) Alternative spelling of Seifert fiber space [(topology, chiefly US) A fiber bundle whose fiber is the circle and whose base space is a two-dimensional orbifold]
n
(topology, Britain) Alternative spelling of Seifert fiber space [(topology, chiefly US) A fiber bundle whose fiber is the circle and whose base space is a two-dimensional orbifold]
adj
(mathematics, of a group G) Having a topology such that, for each c in G, the two functions G → G defined by x↦xc and x↦cx are continuous.
adj
(mathematical analysis, of a topological space) Having a countable dense subset.
n
(mathematical analysis) The property of a topological space that every sequence has a convergent subsequence.
n
(mathematics) An abstract construct in topology that associates data to the open sets of a topological space, together with well-defined restrictions from larger to smaller open sets, subject to the condition that compatible data on overlapping open sets corresponds, via the restrictions, to a unique datum on the union of the open sets.
n
(topology) A two-point Alexandrov-discrete space.
n
(topology) A collection of simplices of various dimensions, connected to one another according to certain rules.
adj
(topology, of a topological space) Having its fundamental group a singleton.
n
A branch of mathematics that studies the properties of topological spaces endowed with ℝᵏ-valued functions, with respect to the change of these functions.
n
(topology) The non-intuitive fact that it is possible to turn a sphere inside out in a three-dimensional space with possible self-intersections but without creating any crease.
n
(topology) A topological space of which every join-irreducible closed subset is the closure of exactly one point of the space.
n
(topology) The topology of a Euclidean space ℝⁿ such that any subset of that space is open (i.e. belonging to the topology) if it can be written as a union of open balls from that space.
n
A physical network topology in which all nodes are connected to a central connectivity device (e.g. a hub).
n
(topology, logic) A topological space constructed from a Boolean algebra, whose points are ultrafilters on the algebra.
n
(mathematics) A collection of linear subspaces of the fibers Vₓ of V at x in X (where V is a vector bundle and X a topological space), that make up a vector bundle in their own right.
n
(topology) Synonym of subcover
n
(topology) A manifold which is a subset of another, so that the inclusion function is an embedding or, sometimes, an immersion.
n
(topology) The abstraction of a sequence.
adj
(mathematical analysis) restricted to an invariant subspace (of a scalar operator)
n
(topology) The topology of a subset S of a topological space X which is obtained by considering any subset of S to be an open set if it corresponds to the intersection of S with some open set of X.
n
(mathematics) subspace topology
adj
(mathematics, topology, of a topological space) Having a subbasis such that every open cover of the topological space from elements of the subbasis has a subcover with at most two subbasis elements.
n
(topology) A topological space derived from another by taking the product of the original space with an interval and collapsing each end of the product to a point.
n
(topology) A fiber bundle for which the base space is a differentiable manifold and each fiber over a point of that manifold is the tangent space of that point.
n
(topology, differential topology) An n-dimensional vector space that represents the set of all vectors tangent to given n-dimensional differentiable manifold M at point x.
n
(mathematics) A space T(S) assigned to a (real) topological (or differential) surface S: it parameterizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism.
adj
(topology) Having the same dimension as the ambient topological space.
adj
(mathematics) Equipped with a topology that is typically required to be compatible with the underlying structure in some way.
n
(mathematics) A field that is also a topological space in which addition and multiplication are both continuous.
n
(topology) A group which is also a topological space and whose group operations are continuous functions.
n
(topology) A property of a topological space that is invariant under homeomorphisms.
n
(graph theory, computer science) An ordering of the vertices of a directed graph such that if an edge goes from vertex u to vertex v then u precedes v in the ordering.
n
(functional analysis) A vector space over a topological field endowed with a topology (often the real or complex numbers with standard topology) such that vector addition and scalar multiplication are continuous functions.
v
(mathematics) To equip with a topology, to characterize as a topological space
adj
(mathematics) Characterized as a topology
n
(computing) The arrangement of nodes in a communications network.
n
(category theory) a Grothendieck topos
n
(topology) A topology consisting only of the empty set and the underlying set.
n
(topology) A topological space which is a product of infinitely many closed intervals.
adj
(of a topological space) Such that, for any two of its points, there exist respective open sets containing each and not the other.
adj
(topology, of a topological space) Such that any two points have disjoint neighborhoods.
n
(topology) A set of "points" upon which a topology is built, thereby yielding a topological space.
adj
(mathematics) Of a topological space X: such that it is a connected space and, for any closed, connected A,B⊂X with X=A∪B, the intersection A∩B is connected.
n
(topology) A space for which properties like uniform continuity and uniform convergence can be formulated.
n
(combinatorics) Uniform spanning tree.
n
(topology, category theory) A fiber bundle for which the fiber is a vector space.
adv
(topology) Of a covering space of finite index.
adj
(of a topological space) Having all its homotopy groups trivial.
n
(topology) A one-point union of a family of topological spaces.
n
(topology) The smallest cardinality of a base.
n
(algebraic geometry) Originally, a topology applicable to algebraic varieties, such that the closed sets are the variety's algebraic subvarieties; later, a generalisation in which the topological space is the set of prime ideals of a commutative ring and is called the spectrum of the ring.
n
(algebraic geometry) A locally ringed space of a subring k of a field K, whose points are valuation rings containing k and contained in K. They generalize the Riemann surface of a complex curve.
adj
(mathematical analysis, of a set in a topological space) That is expressible as a countable union of compact sets.
n
(topology) An order tree whose segments are homeomorphic to segments of the real line and that is the union of countably many segments.
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