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. The idea was to have a class of spaces that was broader than simplicial complexes (we could say now, had better categorical properties); but still retained a combinatorial nature, so that computational considerations were not ignored. The name itself is unrevealing: CW stands for closure-finite weak topology.

Attaching cells

A cell is attached by gluing a closed n-dimensional ball Dn to the (n−1)-skeleton Xn−1, i.e., the union of all lower dimensional cells. The gluing is specified by a continuous function f from ∂Dn = Sn-1 to Xn−1. The points on the new space are exactly the equivalence classes of points in the disjoint union of the old space and the closed cell Dn, the equivalence relation being the transitive closure of x≡f(x). The function f plays an essential role in determining the nature of the newly enlarged complex. For example, if D2 is glued onto S1 in the usual way, we get D2 itself; if f has winding number 2, we get the real projective plane instead.

Related Topics:
Attached - Winding number - Real projective plane

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