# 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.

## 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).