  Take finitely many topological spaces and for each pair of these spaces
choose a pair of corresponding closed subspaces that are identified by a
homeomorpism. We note that this gluing procedure does not guarantee that the
building pieces, or the gluings of some pieces, are embedded in the space
obtained by putting together all given ingredients. Dually, we show that a
certain sufficient condition, called the cocycle condition, is also necessary
to guarantee sheaf-like properties of surjective multi-pullbacks of algebras
with distributive lattices of ideals.
