  We show that a projective space P^\infty(Z/2) endowed with the Alexandrov
topology 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 projective space. In technical terms, we prove that the
category of finitely supported flabby sheaves of algebras is equivalent to the
category of algebras with a finite set of ideals that intersect to zero and
generate a distributive lattice. In particular, the Gelfand transform allows us
to view finite closed coverings of compact Hausdorff spaces as flabby sheaves
of commutative C*-algebras over P^\infty(Z/2).
