We consider the poset of all non-empty finite subsets of the set of natural numbers, use the poset structure to topologise it with the Alexandrov topology, and call the thus obtained topological space ${\mathbb{P}}^{\infty }$ the universal partition space. Then we show that it is a classifying space for finite closed coverings of compact quantum spaces in the sense that any such a covering is functorially equivalent to a sheaf over this partition space. In technical terms, we prove that the category of finitely supported...

We prove the Finsler analog of the conformal Lichnerowicz-Obata conjecture showing that a complete and essential conformal vector field on a non-Riemannian Finsler manifold is a homothetic vector field of a Minkowski metric.

Let T ∈ L(E)ⁿ be a commuting tuple of bounded linear operators on a complex Banach space E and let ${\sigma }_F\left(T\right)=\sigma \left(T\right)\setminus {\sigma }_e\left(T\right)$ be the non-essential spectrum of T. We show that, for each connected component M of the manifold $Reg\left({\sigma }_F\left(T\right)\right)$ of all smooth points of ${\sigma }_F\left(T\right)$, there is a number p ∈ 0, ..., n such that, for each point z ∈ M, the dimensions of the cohomology groups $H^p({(z-T)}^k,E)$ grow at least like the sequence ${\left(k^d\right)}_{k\ge 1}$ with d = dim M.