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IMI/Publicaţii/BASM/Ediţii/BASM n.2 (66), 2011/

Estimation of the number of one-point expansions of a topology which is given on a finite set

Authors: V. I. Arnautov

Abstract

Let X be a finite set and τ be a topology on X which has precisely m open sets. If t(τ) is the number of possible one-point expansions of the topology τ on Y=X∪{y}, then ({m • (m+3)} / {2})-1 ≥ t(τ) ≥ 2 • m+log2m-1 and ({m • (m+3)} / {2})-1 = t(τ) if and only if τ is a chain (i.e. it is a linearly ordered set) and t(τ) =2 • m+ log2m-1 if and only if τ is an atomistic lattice.

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