# Cellular induction

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Cellular induction is a technique used to prove things about cellular spaces, particularly about CW-complexes, and about their underlying topological spaces.

## Contents

## Overall proof scheme

Most proofs using cellular induction require us to construct a particular object for the -skeleton in a naturally compatible manner. The proof thus has the following ingredients:

- Constructing an initial object (i.e. for the -skeleton)
- Showing how to pass from the object constructed for the -skeleton to an object constructed for the -skeleton, in a compatible way
- Arguing that this gives the required object for the whole space

## The induction step

The induction step is usually the step where the maximum hardwork needs to be done. The induction step usually consists of two parts:

- One part involves reducing the problem to the case where we are attaching only one cell. This step usually invokes the fact that the interiors of the -cells attached do
*not*interfer with each other, and so we can independently prove things for each -cell attached. - The second part involves actually proving the result for attaching a single cell.

### Type of facts used in the induction step

In this step, we need to use the topological properties of the -disc, and its boundary. In particular, the -disc is compact, normal, metrizable, contractible, path-connected, locally path-connected. Also the boundary is a strong deformation retract of the disc minus any point in the interior, and the boundary is a closed subset of the disc.

## The final step

The final step uses two kinds of arguments:

- The fact that the topology on the whole space is completely determined by what happens at -skeletons
- The fact that any compact subset must lie completely inside a -skeleton for some finite (used particularly when dealing with CW-complexes)

### Proofs involving the construction of subsets

The key fact to use here is that a subset is open (respectively, closed) in if and only if is open (respectively closed) in . Thus, if our goal is to construct an open or closed set satisfying certain properties, we can construct an ascending sequence of open (respectively, closed) subsets, with the property that for . The union of these gives an open (respectively, closed) set.

Of course, some more work may be needed to show that the way we choose s ensures that their union has the desired properties.

### Proofs involving the construction of maps or homotopies

The key fact here is that if is a function such that is continuous for every , then so is . Thus, to obtain a continuous function , we can work by obtaining a sequence such that for . Then, the naturally defined which sends to for any with is continuous.

We can do the same thing for homotopies between maps from to ; this is a crucial ingredient in the proof of the cellular homotopy theorem (also called cellular approximation theorem).