Cellular induction - Revision history

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2008-05-11T19:40:21Z

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Vipul at 18:37, 27 October 2007

2007-10-27T18:37:43Z

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	Cellular induction is a technique used to prove things about [[cellular space]]s, particularly about [[CW-complex]]es, and about their underlying topological spaces.		Cellular induction is a technique used to prove things about [[cellular space]]s, particularly about [[CW-complex]]es, and about their underlying topological spaces.

	==~~Constructive proofs using cellular induction~~==		==Overall proof scheme==

	Most proofs using cellular induction require us to construct a particular object for the <math>n</math>-skeleton in a naturally compatible manner. The proof thus has the following ingredients:		Most proofs using cellular induction require us to construct a particular object for the <math>n</math>-skeleton in a naturally compatible manner. The proof thus has the following ingredients:
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	* Arguing that this gives the required object for the whole space		* 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 <math>n</math>-cells attached do ''not'' interfer with each other, and so we can independently prove things for each <math>n</math>-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 <math>n</math>-disc, and its boundary. In particular, the <math>n</math>-disc is [[compact space\|compact]], [[normal space\|normal]], [[metrizable space\|metrizable]], [[contractible space\|contractible]], [[path-connected space\|path-connected]], [[locally path-connected space\|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 final step uses two kinds of arguments:

Vipul at 00:20, 27 October 2007

2007-10-27T00:20:38Z

New page

{{survey article}}

Cellular induction is a technique used to prove things about [[cellular space]]s, particularly about [[CW-complex]]es, 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 <math>n</math>-skeleton in a naturally compatible manner. The proof thus has the following ingredients:

* Constructing an initial object (i.e. for the <math>-1></math>-skeleton)
* Showing how to pass from the object constructed for the <math>n-1</math>-skeleton to an object constructed for the <math>n</math>-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 <math>n</math>-skeletons
* The fact that any compact subset must lie completely inside a <math>n</math>-skeleton for some finite <math>n</math> (used particularly when dealing with CW-complexes)

===Proofs involving the construction of subsets===

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

===Proofs involving the construction of maps or homotopies===

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

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