  Extending work of Budzynski and Kondracki, we investigate coverings and
gluings of algebras and differential algebras. We describe in detail the gluing
of two quantum discs along their classical subspace, giving a C*-algebra
isomorphic to a certain Podles sphere, as well as the gluing of
U_{\sqrt{q}}(sl_2)-covariant differential calculi on the discs.
