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. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial naturethat allows for computation (often with a much smaller 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. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial naturethat allows for computation (often with a much smaller complex). Roughly speaking, a CW complex is made of basic building blocks called cells. The precise definition prescribes how the cells may be topologically glued together. The C stands for 'closure-finite', and the W for 'weak' topology. An n-dimensional closed cell is the image of an n-dimensional closed ball under an attaching map. For example, a simplex is a closed cell, and more generally, a convex polytope is a closed cell. An n-dimensional open cell is a topological space that is homeomorphic to the n-dimensional open ball. A 0-dimensional open (and closed) cell is a singleton space. Closure-finite means that each closed cell is covered by a finite union of open cells (or meets only finitely many other cells). A CW complex is a Hausdorff space X together with a partition of X into open cells (of perhaps varying dimension) that satisfies two additional properties: A CW complex is called regular if for each n-dimensional open cell C in the partition of X, the continuous map f from the n-dimensional closed ball to X is a homeomorphism onto the closure of the cell C. Roughly speaking, a relative CW complex differs from a CW complex in that we allow it to have one extra building block which does not necessarily possess a cellular structure. This extra-block can be treated as a (-1)-dimensional cell in the former definition. If the largest dimension of any of the cells is n, then the CW complex is said to have dimension n. If there is no bound to the cell dimensions then it is said to be infinite-dimensional. The n-skeleton of a CW complex is the union of the cells whose dimension is at most n. If the union of a set of cells is closed, then this union is itself a CW complex, called a subcomplex. Thus the n-skeleton is the largest subcomplex of dimension n or less. A CW complex is often constructed by defining its skeleta inductively by 'attaching' cells of increasing dimension.By an 'attachment' of an n-cell to a topological space X one means an adjunction space B ∪ f X {displaystyle Bcup _{f}X} where f is a continuous map from the boundary of a closed n-dimensional ball B ⊂ R n {displaystyle Bsubset R^{n}} to X. To construct a CW complex, begin with a 0-dimensional CW complex, that is, a discrete space X 0 {displaystyle X_{0}} . Attach 1-cells to X 0 {displaystyle X_{0}} to obtain a 1-dimensional CW complex X 1 {displaystyle X_{1}} . Attach 2-cells to X 1 {displaystyle X_{1}} to obtain a 2-dimensional CW complex X 2 {displaystyle X_{2}} . Continuing in this way, we obtain a nested sequence of CW complexes X 0 ⊂ X 1 ⊂ ⋯ X n ⊂ ⋯ {displaystyle X_{0}subset X_{1}subset cdots X_{n}subset cdots } of increasing dimension such that if i ≤ j {displaystyle ileq j} then X i {displaystyle X_{i}} is the i-skeleton of X j {displaystyle X_{j}} . Up to isomorphism every n-dimensional CW complex can be obtained from its (n − 1)-skeleton via attaching n-cells, and thus every finite-dimensional CW complex can be built up by the process above. This is true even for infinite-dimensional complexes, with the understanding that the result of the infinite process is the direct limit of the skeleta: a set is closed in X if and only if it meets each skeleton in a closed set.