A Euler-Poincaré characteristic for the open set lattices of finite topologies |
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Authors: | Shawpawn Kumar Das |
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Institution: | 10, Raja Dinendra Street, Calcutta 700 009, India |
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Abstract: | Let be a finite topology. If P and Q are open sets of (Q may be the null set) then P is a minimal cover of Q provided Q ? P and there does not exist any open set R of such that Q ? R ? P. A subcollection of the open sets of is termed an i-discrete collection of provided contains every open O ∈ with the property that ? ? O ? ? , contains exactly i minimal covers of ? , and provided ? = ?{O | O ∈ and O is a minimal cover of ? }. A single open set is a O-discrete collection. The number of distinct i-discrete collections of is denoted by p(, i). If there does not exist any i-discrete collection then p(,i) = 0, and this happens trivially for the case when i is greater than the number of points on which is defined. The object of this article is to establish the theorem: For any finite topology , the quantity E() = Σi = 0∞ (?1)ip(, i) = 1. |
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