INCLUSION-EXCLUSION COMPLEXES FOR PSEUDODISK COLLECTIONS

Let B be a finite pseudodisk collection in the plane. By the principle of inclusion-exclusion, the area or any other measure of the union is [GRAPHICS] We show the existence of a two-dimensional abstract simpli cial complex, X subset of or equal to 2(B), so the above relation hold s even if X is substituted for 2(B). In addition, X can be embedded in R(2) SO its underlying space is homotopy equivalent to int Boolean OR B, and the frontier of X is isomorphic to the nerve of the set of bou ndary contributions.