Motivated by the recent surge of interest in the concept of injective C*-algebras, we return to an old result of Gleason characterizing the projective cover of a given compact topological space. Here we go in a different direction and attempt to find all compact spaces having a given projective resolution and we discover that this is intimately related to what we call an \emph{orthogonality relation} in the complete Boolean algebra of all regular open sets of a topological space.