We say that the locally
countable sum theorem holds for a property 𝒫 if whenever {Fα: α ∈ Λ} is a locally
countable closed covering of X such that each Fα has 𝒫, then X has 𝒫. We prove
some general theorems which establish the locally countable sum theorem for
properties satisfying certain conditions. It is then shown that a number of sum
theorems hold for those properties which are closed hereditary and for which the
locally countable sum theorem holds. We apply our theorems to some particular
cases to obtain many new results and also to improve upon some known
results.
