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).