By a sum theorem for
topological spaces is meant a theorem of the following type: If ℱ is a cover of a space
X, each element of which possesses a property 𝒫, then X also possesses the
property 𝒫. Different types of sum theorems for various classes of topological
spaces have been obtained from time to time by various authors. Perhaps the
simplest known sum theorem is the locally finite sum theorem which states the
following: If {Fα;α ∈ Λ} be a locally finite closed covering of a space X such
that each Fα possesses a property 𝒫, then X possesses 𝒫 The locally finite
sum theorem is shown to hold for a large number of important topological
properties.
