Cellular induction

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

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


 * Constructing an initial object (i.e. for the $$-1>$$-skeleton)
 * Showing how to pass from the object constructed for the $$n-1$$-skeleton to an object constructed for the $$n$$-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 $$n$$-cells attached do not interfer with each other, and so we can independently prove things for each $$n$$-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 $$n$$-disc, and its boundary. In particular, the $$n$$-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 $$n$$-skeletons
 * The fact that any compact subset must lie completely inside a $$n$$-skeleton for some finite $$n$$ (used particularly when dealing with CW-complexes)

Proofs involving the construction of subsets
The key fact to use here is that a subset $$U \subset X$$ is open (respectively, closed) in $$X$$ if and only if $$U \cap X^n$$ is open (respectively closed) in $$X^n$$. Thus, if our goal is to construct an open or closed set satisfying certain properties, we can construct an ascending sequence $$U_n$$ of open (respectively, closed) subsets, with the property that $$U_n \cap X^m = U_m$$ for $$m < n$$. 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 $$U_n$$s ensures that their union has the desired properties.

Proofs involving the construction of maps or homotopies
The key fact here is that if $$f:X \to Y$$ is a function such that $$f|_{X^n}$$ is continuous for every $$n$$, then so is $$f$$. Thus, to obtain a continuous function $$f:X \to Y$$, we can work by obtaining a sequence $$f_n:X^n \to Y$$ such that $$f_n|_{X^m} = f_m$$ for $$m < n$$. Then, the naturally defined $$f$$ which sends $$x$$ to $$f_n(x)$$ for any $$n$$ with $$x \in X^n$$ is continuous.

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