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The ramification of a polyhedral space is defined as the metric completion of the universal cover of its regular locus. We consider mainly polyhedral spaces of two origins: quotients of Euclidean space by a discrete group of isometries and polyhedral metrics on 
$\mathbb{C}\text{P}^{2}$ with singularities at a collection of complex lines. In the former case we conjecture that quotient spaces always have a 
$\text{CAT}[0]$ ramification and prove this in several cases. In the latter case we prove that the ramification is 
$\text{CAT}[0]$ if the metric on 
$\mathbb{C}\text{P}^{2}$ is non-negatively curved. We deduce that complex line arrangements in 
$\mathbb{C}\text{P}^{2}$ studied by Hirzebruch have aspherical complement.
We prove that, given any covering of any infinite-dimensional Hilbert space 
$H$ by countably many closed balls, some point exists in $H$ which belongs to infinitely many balls. We do that by characterizing isomorphically polyhedral separable Banach spaces as those whose unit sphere admits a point-finite covering by the union of countably many slices of the unit ball.
