**Cellular Approximation**

In algebraic topology, in the **cellular approximation theorem**, a map between CW-complexes can always be taken to be of a specific type. Concretely, if *X* and *Y* are CW-complexes, and *f* : *X* → *Y* is a continuous map, then *f* is said to be *cellular*, if *f* takes the *n*-skeleton of *X* to the *n*-skeleton of *Y* for all *n*, i.e. if for all *n*. The content of the cellular approximation theorem is then that any continuous map *f* : *X* → *Y* between CW-complexes *X* and *Y* is homotopic to a cellular map, and if *f* is already cellular on a subcomplex *A* of *X*, then we can furthermore choose the homotopy to be stationary on *A*. From an algebraic topological viewpoint, any map between CW-complexes can thus be taken to be cellular.

Read more about Cellular Approximation: Idea of Proof

### Other articles related to "cellular approximation, cellular":

**Cellular Approximation**For Pairs

... Then f is homotopic to a

**cellular**map (X,A)→(Y,B) ... To see this, restrict f to A and use

**cellular approximation**to obtain a homotopy of f to a

**cellular**map on A ... extend this homotopy to all of X, and apply

**cellular approximation**again to obtain a map

**cellular**on X, but without violating the

**cellular**property on A ...