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.

Constructive proofs using cellular induction

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 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).