, Volume 17, Issue 3, pp 287-306

Inclusion-exclusion complexes for pseudodisk collections

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Abstract

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

$$\mu \left( { \cup B} \right) = \sum\limits_{\sigma \in 2^B - \left\{ {\not 0} \right\}} {( - 1)^{card \sigma - 1} \mu \left( { \cap \sigma } \right)} .$$
.

We show the existence of a two-dimensional abstract simplicial complex, χ ⊆ 2 B , so the above relation holds even if χ is substituted for 2B. In addition, χ can be embedded in ℝ2 so its underlying space is homotopy equivalent to int ⋃ B, and the frontier of χ is isomorphic to the nerve of the set of boundary contributions.