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Concept# CW complex

Summary

A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better properties than simplicial complexes, but still retains a combinatorial nature that allows for computation (often with a much smaller complex). The C stands for "closure-finite", and the W for "weak" topology.
A CW complex is constructed by taking the union of a sequence of topological spacessuch that each is obtained from by gluing copies of k-cells , each homeomorphic to , to by continuous gluing maps . The maps are also called attaching maps.
Each is called the k-skeleton of the complex.
The topology of is weak topology: a subset is open iff is open for each cell .
In the language of category theory, the topology on is the direct limit of the diagram The name "CW" stands for "closure-finite weak topology", which is explained by the following theorem:
This partition of X is also called a cellulation.
The CW complex construction is a straightforward generalization of the following process:
A 0-dimensional CW complex is just a set of zero or more discrete points (with the discrete topology).
A 1-dimensional CW complex is constructed by taking the disjoint union of a 0-dimensional CW complex with one or more copies of the unit interval. For each copy, there is a map that "glues" its boundary (its two endpoints) to elements of the 0-dimensional complex (the points). The topology of the CW complex is the topology of the quotient space defined by these gluing maps.
In general, an n-dimensional CW complex is constructed by taking the disjoint union of a k-dimensional CW complex (for some ) with one or more copies of the n-dimensional ball. For each copy, there is a map that "glues" its boundary (the -dimensional sphere) to elements of the -dimensional complex. The topology of the CW complex is the quotient topology defined by these gluing maps.

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CW complex

A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better properties than simplicial complexes, but still retains a combinatorial nature that allows for computation (often with a much smaller complex). The C stands for "closure-finite", and the W for "weak" topology.

Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an -dimensional manifold, or -manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of -dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane.

Simplicial complex

In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their n-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. To distinguish a simplicial complex from an abstract simplicial complex, the former is often called a geometric simplicial complex.