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CW complex

Index CW complex

In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. [1]

67 relations: Adjunction space, Alexandroff extension, Algebraic variety, Allen Hatcher, Atiyah–Hirzebruch spectral sequence, Baire space, Ball (mathematics), Boundary (topology), Brown's representability theorem, Cambridge University Press, Cartesian product, Category theory, Cellular homology, Chain complex, Closed set, Compact space, Compact-open topology, Compactly generated space, Comparison of topologies, Convex polytope, Cover (topology), Covering space, Cubic graph, Differentiable manifold, Direct limit, Discrete space, Discrete two-point space, European Mathematical Society, Function space, Generic property, Graph (discrete mathematics), Grassmannian, Handle decomposition, Hausdorff space, Hawaiian earring, Hilbert space, Homeomorphism, Homology (mathematics), Homotopy, Homotopy category, Hyperbolic manifold, Integer lattice, J. H. C. Whitehead, John Milnor, Locally compact space, N-skeleton, N-sphere, Paracompact space, Partition of a set, Pointed space, ..., Polyhedron, Presentation of a group, Product topology, Projective space, Representable functor, Simplex, Simplicial complex, Singleton (mathematics), Singular homology, SnapPea, Surgery theory, Tietze transformations, Topological space, Topology, Tree (graph theory), Van Nostrand, Whitehead theorem. Expand index (17 more) »

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.

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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.

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Algebraic variety

Algebraic varieties are the central objects of study in algebraic geometry.

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Allen Hatcher

Allen Edward Hatcher (born October 23, 1944) is an American topologist.

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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.

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Baire space

In mathematics, a Baire space is a topological space such that every intersection of a countable collection of open dense sets in the space is also dense.

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Ball (mathematics)

In mathematics, a ball is the space bounded by a sphere.

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Boundary (topology)

In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. The term boundary operation refers to finding or taking the boundary of a set.

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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.

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Cambridge University Press

Cambridge University Press (CUP) is the publishing business of the University of Cambridge.

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Cartesian product

In set theory (and, usually, in other parts of mathematics), a Cartesian product is a mathematical operation that returns a set (or product set or simply product) from multiple sets.

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Category theory

Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms).

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Cellular homology

In mathematics, cellular homology in algebraic topology is a homology theory for the category of CW-complexes.

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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.

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Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.

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Compact space

In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).

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Compact-open topology

In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces.

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Compactly generated space

In topology, a compactly generated space (or k-space) is a topological space whose topology is coherent with the family of all compact subspaces.

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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.

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Convex polytope

A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn.

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Cover (topology)

In mathematics, a cover of a set X is a collection of sets whose union contains X as a subset.

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Covering space

In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below.

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Cubic graph

In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three.

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Differentiable manifold

In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus.

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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.

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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 discontinuous sequence, meaning they are isolated from each other in a certain sense.

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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.

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European Mathematical Society

The European Mathematical Society (EMS) is a European organization dedicated to the development of mathematics in Europe.

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Function space

In mathematics, a function space is a set of functions between two fixed sets.

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Generic property

In mathematics, properties that hold for "typical" examples are called generic properties.

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Graph (discrete mathematics)

In mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related".

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Grassmannian

In mathematics, the Grassmannian is a space which parametrizes all -dimensional linear subspaces of the n-dimensional vector space.

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Handle decomposition

In mathematics, a handle decomposition of an m-manifold M is a union where each M_i is obtained from M_ by the attaching of i-handles.

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Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods.

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Hawaiian earring

In mathematics, the Hawaiian earring H is the topological space defined by the union of circles in the Euclidean plane R2 with center (0) and radius for.

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Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.

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Homeomorphism

In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function.

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Homology (mathematics)

In mathematics, homology is a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces.

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Homotopy

In topology, two continuous functions from one topological space to another are called homotopic (from Greek ὁμός homós "same, similar" and τόπος tópos "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.

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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.

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Hyperbolic manifold

In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension.

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Integer lattice

In mathematics, the n-dimensional integer lattice (or cubic lattice), denoted Zn, is the lattice in the Euclidean space Rn whose lattice points are ''n''-tuples of integers.

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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.

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John Milnor

John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, K-theory and dynamical systems.

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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.

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N-skeleton

In mathematics, particularly in algebraic topology, the of a topological space X presented as a simplicial complex (resp. CW complex) refers to the subspace Xn that is the union of the simplices of X (resp. cells of X) of dimensions In other words, given an inductive definition of a complex, the is obtained by stopping at the.

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N-sphere

In mathematics, the n-sphere is the generalization of the ordinary sphere to spaces of arbitrary dimension.

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Paracompact space

In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite.

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Partition of a set

In mathematics, a partition of a set is a grouping of the set's elements into non-empty subsets, in such a way that every element is included in one and only one of the subsets.

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Pointed space

In mathematics, a pointed space is a topological space with a distinguished point, the basepoint.

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Polyhedron

In geometry, a polyhedron (plural polyhedra or polyhedrons) is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices.

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Presentation of a group

In mathematics, one method of defining a group is by a presentation.

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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.

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Projective space

In mathematics, a projective space can be thought of as the set of lines through the origin of a vector space V. The cases when and are the real projective line and the real projective plane, respectively, where R denotes the field of real numbers, R2 denotes ordered pairs of real numbers, and R3 denotes ordered triplets of real numbers.

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Representable functor

In mathematics, particularly category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets.

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Simplex

In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions.

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Simplicial complex

In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration).

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Singleton (mathematics)

In mathematics, a singleton, also known as a unit set, is a set with exactly one element.

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Singular homology

In algebraic topology, a branch of mathematics, 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).

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SnapPea

SnapPea is free software designed to help mathematicians, in particular low-dimensional topologists, study hyperbolic 3-manifolds.

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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.

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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.

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Topological space

In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

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Topology

In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.

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Tree (graph theory)

In mathematics, and more specifically in graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one path.

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Van Nostrand

Van Nostrand is a surname.

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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.

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Attaching a cell, CW Complex, CW complexes, CW pair, CW-Complex, CW-complex, CW-complexes, CW-pair, CW-structure, Category of CW-complexes, Cell complex, Cellular complex, Closure-finite, Cw complex, Cw-complex.

References

[1] https://en.wikipedia.org/wiki/CW_complex

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