CW complex, the Glossary
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology.[1]
Table of Contents
78 relations: Abstract cell complex, Adjunction space, Alexandroff extension, Algebraic topology, Algebraic variety, Allen Hatcher, Atiyah–Hirzebruch spectral sequence, Baire space, Ball (mathematics), Brown's representability theorem, Bulletin of the American Mathematical Society, Cambridge University Press, Cartesian product, Category theory, Cellular homology, Chain complex, Closed set, Compact space, Compact-open topology, Compactly generated space, Comparison of topologies, Covering space, Cubic graph, David Van Nostrand, Differentiable manifold, Direct limit, Discrete space, Discrete two-point space, Disjoint union (topology), Equator, European Mathematical Society, Function space, Generic property, Graph (discrete mathematics), Graph embedding, Grassmannian, Handle decomposition, Hausdorff space, Hawaiian earring, Hedgehog space, Hilbert space, Homeomorphism, Homology (mathematics), Homotopical connectivity, Homotopy, Homotopy category, Homotopy theory, Hyperbolic manifold, If and only if, Integer lattice, ... Expand index (28 more) »
Abstract cell complex
In mathematics, an abstract cell complex is an abstract set with Alexandrov topology in which a non-negative integer number called dimension is assigned to each point.
See CW complex and Abstract cell complex
Adjunction space
In mathematics, an adjunction space (or attaching space) is a common construction in topology where one topological space is attached or "glued" onto another. CW complex and adjunction space are topological spaces.
See CW complex and Adjunction space
Alexandroff extension
In the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact.
See CW complex and Alexandroff extension
Algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces.
See CW complex and Algebraic topology
Algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics.
See CW complex and Algebraic variety
Allen Hatcher
Allen Edward Hatcher (born October 23, 1944) is an American topologist.
See CW complex and Allen Hatcher
Atiyah–Hirzebruch spectral sequence
In mathematics, the Atiyah–Hirzebruch spectral sequence is a spectral sequence for calculating generalized cohomology, introduced by in the special case of topological K-theory.
See CW complex and Atiyah–Hirzebruch spectral sequence
Baire space
In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior.
See CW complex and Baire space
Ball (mathematics)
In mathematics, a ball is the solid figure bounded by a sphere; it is also called a solid sphere.
See CW complex and Ball (mathematics)
Brown's representability theorem
In mathematics, Brown's representability theorem in homotopy theory gives necessary and sufficient conditions for a contravariant functor F on the homotopy category Hotc of pointed connected CW complexes, to the category of sets Set, to be a representable functor.
See CW complex and Brown's representability theorem
Bulletin of the American Mathematical Society
The Bulletin of the American Mathematical Society is a quarterly mathematical journal published by the American Mathematical Society.
See CW complex and Bulletin of the American Mathematical Society
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge.
See CW complex and Cambridge University Press
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets and, denoted, is the set of all ordered pairs where is in and is in.
See CW complex and Cartesian product
Category theory
Category theory is a general theory of mathematical structures and their relations.
See CW complex and Category theory
Cellular homology
In mathematics, cellular homology in algebraic topology is a homology theory for the category of CW-complexes.
See CW complex and Cellular homology
Chain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of the next.
See CW complex and Chain complex
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.
Compact space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space.
See CW complex and Compact space
Compact-open topology
In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces.
See CW complex and Compact-open topology
Compactly generated space
In topology, a topological space X is called a compactly generated space or k-space if its topology is determined by compact spaces in a manner made precise below. CW complex and compactly generated space are homotopy theory.
See CW complex and Compactly generated space
Comparison of topologies
In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set.
See CW complex and Comparison of topologies
Covering space
In topology, a covering or covering projection is a map between topological spaces that, intuitively, locally acts like a projection of multiple copies of a space onto itself. CW complex and covering space are algebraic topology and homotopy theory.
See CW complex and Covering space
Cubic graph
In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three.
See CW complex and Cubic graph
David Van Nostrand
David Van Nostrand (December 5, 1811 – June 14, 1886) was a New York City publisher.
See CW complex and David Van Nostrand
Differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus.
See CW complex and Differentiable manifold
Direct limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way.
See CW complex and Direct limit
Discrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a, meaning they are isolated from each other in a certain sense. CW complex and discrete space are topological spaces.
See CW complex and Discrete space
Discrete two-point space
In topology, a branch of mathematics, a discrete two-point space is the simplest example of a totally disconnected discrete space. CW complex and discrete two-point space are topological spaces.
See CW complex and Discrete two-point space
Disjoint union (topology)
In general topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, free sum, topological sum, or coproduct) of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology.
See CW complex and Disjoint union (topology)
Equator
The equator is a circle of latitude that divides a spheroid, such as Earth, into the Northern and Southern hemispheres.
European Mathematical Society
The European Mathematical Society (EMS) is a European organization dedicated to the development of mathematics in Europe.
See CW complex and European Mathematical Society
Function space
In mathematics, a function space is a set of functions between two fixed sets.
See CW complex and Function space
Generic property
In mathematics, properties that hold for "typical" examples are called generic properties.
See CW complex and Generic property
Graph (discrete mathematics)
In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a set of objects where some pairs of the objects are in some sense "related".
See CW complex and Graph (discrete mathematics)
Graph embedding
In topological graph theory, an embedding (also spelled imbedding) of a graph G on a surface \Sigma is a representation of G on \Sigma in which points of \Sigma are associated with vertices and simple arcs (homeomorphic images of) are associated with edges in such a way that.
See CW complex and Graph embedding
Grassmannian
In mathematics, the Grassmannian \mathbf_k(V) (named in honour of Hermann Grassmann) is a differentiable manifold that parameterizes the set of all k-dimensional linear subspaces of an n-dimensional vector space V over a field K. For example, the Grassmannian \mathbf_1(V) is the space of lines through the origin in V, so it is the same as the projective space \mathbf(V) of one dimension lower than V.
See CW complex and Grassmannian
Handle decomposition
In mathematics, a handle decomposition of an m-manifold M is a union \emptyset.
See CW complex and Handle decomposition
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each that are disjoint from each other.
See CW complex and Hausdorff space
Hawaiian earring
In mathematics, the Hawaiian earring \mathbb is the topological space defined by the union of circles in the Euclidean plane \R^2 with center \left(\tfrac,0\right) and radius \tfrac for n. CW complex and Hawaiian earring are topological spaces.
See CW complex and Hawaiian earring
Hedgehog space
In mathematics, a hedgehog space is a topological space consisting of a set of spines joined at a point. CW complex and hedgehog space are topological spaces.
See CW complex and Hedgehog space
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional.
See CW complex and Hilbert space
Homeomorphism
In mathematics and more specifically in topology, a homeomorphism (from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function.
See CW complex and Homeomorphism
Homology (mathematics)
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces.
See CW complex and Homology (mathematics)
Homotopical connectivity
In algebraic topology, homotopical connectivity is a property describing a topological space based on the dimension of its holes. CW complex and homotopical connectivity are homotopy theory.
See CW complex and Homotopical connectivity
Homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from ὁμός "same, similar" and τόπος "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions. CW complex and homotopy are homotopy theory.
Homotopy category
In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. CW complex and homotopy category are homotopy theory.
See CW complex and Homotopy category
Homotopy theory
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. CW complex and homotopy theory are algebraic topology.
See CW complex and Homotopy theory
Hyperbolic manifold
In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension.
See CW complex and Hyperbolic manifold
If and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements.
See CW complex and If and only if
Integer lattice
In mathematics, the -dimensional integer lattice (or cubic lattice), denoted, is the lattice in the Euclidean space whose lattice points are n-tuples of integers.
See CW complex and Integer lattice
J. H. C. Whitehead
John Henry Constantine Whitehead FRS (11 November 1904 – 8 May 1960), known as "Henry", was a British mathematician and was one of the founders of homotopy theory.
See CW complex and J. H. C. Whitehead
John Milnor
John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems.
See CW complex and John Milnor
K-vertex-connected graph
In graph theory, a connected graph is said to be -vertex-connected (or -connected) if it has more than vertices and remains connected whenever fewer than vertices are removed.
See CW complex and K-vertex-connected graph
Locally compact space
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.
See CW complex and Locally compact space
Loop (graph theory)
In graph theory, a loop (also called a self-loop or a buckle) is an edge that connects a vertex to itself.
See CW complex and Loop (graph theory)
Multigraph
In mathematics, and more specifically in graph theory, a multigraph is a graph which is permitted to have multiple edges (also called parallel edges), that is, edges that have the same end nodes.
N-sphere
In mathematics, an -sphere or hypersphere is an -dimensional generalization of the -dimensional circle and -dimensional sphere to any non-negative integer.
Paracompact space
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite.
See CW complex and Paracompact space
Partition of a set
In mathematics, a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset.
See CW complex and Partition of a set
Pointed space
In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. CW complex and pointed space are homotopy theory and topological spaces.
See CW complex and Pointed space
Polyhedron
In geometry, a polyhedron (polyhedra or polyhedrons) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
Presentation of a group
In mathematics, a presentation is one method of specifying a group.
See CW complex and Presentation of a group
Product topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology.
See CW complex and Product topology
Projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet at infinity.
See CW complex and Projective space
Quotient space (topology)
In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient topology, that is, with the finest topology that makes continuous the canonical projection map (the function that maps points to their equivalence classes).
See CW complex and Quotient space (topology)
Representable functor
In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets.
See CW complex and Representable functor
Simplicial complex
In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). CW complex and simplicial complex are algebraic topology and topological spaces.
See CW complex and Simplicial complex
Singular homology
In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups H_n(X).
See CW complex and Singular homology
SnapPea
SnapPea is free software designed to help mathematicians, in particular low-dimensional topologists, study hyperbolic 3-manifolds.
Surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space.
Surgery theory
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by. CW complex and surgery theory are algebraic topology and homotopy theory.
See CW complex and Surgery theory
Tietze transformations
In group theory, Tietze transformations are used to transform a given presentation of a group into another, often simpler presentation of the same group.
See CW complex and Tietze transformations
Topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. CW complex and topological space are topological spaces.
See CW complex and Topological space
Tree (graph theory)
In graph theory, a tree is an undirected graph in which any two vertices are connected by path, or equivalently a connected acyclic undirected graph.
See CW complex and Tree (graph theory)
Unit interval
In mathematics, the unit interval is the closed interval, that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1.
See CW complex and Unit interval
Weak topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space.
See CW complex and Weak topology
Whitehead theorem
In homotopy theory (a branch of mathematics), the Whitehead theorem states that if a continuous mapping f between CW complexes X and Y induces isomorphisms on all homotopy groups, then f is a homotopy equivalence.
See CW complex and Whitehead theorem
3-sphere
In mathematics, a 3-sphere, glome or hypersphere is a higher-dimensional analogue of a sphere. CW complex and 3-sphere are algebraic topology.
References
[1] https://en.wikipedia.org/wiki/CW_complex
Also known as Attaching a cell, CW complexes, CW pair, CW-Complex, CW-complexes, CW-pair, CW-structure, Category of CW-complexes, Cell complex, Cellular complex, Closure-finite.
, J. H. C. Whitehead, John Milnor, K-vertex-connected graph, Locally compact space, Loop (graph theory), Multigraph, N-sphere, Paracompact space, Partition of a set, Pointed space, Polyhedron, Presentation of a group, Product topology, Projective space, Quotient space (topology), Representable functor, Simplicial complex, Singular homology, SnapPea, Surface, Surgery theory, Tietze transformations, Topological space, Tree (graph theory), Unit interval, Weak topology, Whitehead theorem, 3-sphere.