Dense subspaces and addition theorems for paracompactness

We establish that if a normal space X is the union of a finite collection of dense paracompact subspaces, then X is paracompact (Theorem 2.1), and prove that if a space X is the union of a finite family of dense paracompact subspaces, then X is metacompact (Theorem 2.2). A new class of strongly metacompact spaces is introduced and applied. In particular, it behaves nicely under taking the union of a finite collection of dense subspaces.