CW complex
Introduction
A CW complex is a type of topological space introduced by J.H.C. Whitehead to meet the needs of homotopy theory. This class of spaces is named for the two mathematicians who developed their theory: Clifford Henry Taubes and J.H.C. Whitehead.
Definition
In topology, a branch of mathematics, a CW complex is a type of 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 named for the two mathematicians who developed their theory: Clifford Henry Taubes and J.H.C. Whitehead. The 'C' stands for 'closure-finite', and the 'W' for 'weak' topology.
Construction
A CW complex is built up from simpler ones by attaching cells. The precise definition of a CW complex is technical and involves the notion of a weak topology. The idea is to start with a discrete set (which is a trivially a topological space), and then to "build up" higher-dimensional spaces by attaching cells in a "nice" way.
Properties
CW complexes have many properties that make them well-suited for algebraic topology. For example, every CW complex is a Hausdorff space and a regular space. Furthermore, the weak topology on a CW complex coincides with the quotient topology and the cell topology.
Applications
CW complexes are used in many areas of mathematics, particularly in algebraic topology. They provide a convenient framework for homology theory, cohomology theory, and the theory of characteristic classes. They are also used in the study of manifolds and orbifolds.