Editing Codimension (section)

==Additivity of codimension and dimension counting==
The fundamental property of codimension lies in its relation to  [[intersection (set theory)|intersection]]: if ''W''<sub>1</sub> has codimension ''k''<sub>1</sub>, and ''W''<sub>2</sub> has codimension ''k''<sub>2</sub>, then if ''U'' is their intersection with codimension ''j'' we have

:max (''k''<sub>1</sub>, ''k''<sub>2</sub>) &le; ''j'' &le; ''k''<sub>1</sub> + ''k''<sub>2</sub>.

In fact ''j'' may take any [[integer]] value in this range. This statement is more perspicuous than the translation in terms of dimensions, because the [[Sides of an equation|RHS]] is just the sum of the codimensions. In words

:''codimensions (at most) add''.
:If the subspaces or submanifolds intersect [[Transversality (mathematics)|transversally]] (which occurs [[General position|generically]]), codimensions add exactly.

This statement is called '''dimension counting,''' particularly in [[intersection theory]].