ABSTRACT
Partitions of unity play an important role in the existence of continuous selections of some lower semicontinuous classes and in approximation of upper multivalued maps.
Definition 4.1. Let {Uλ : λ ∈ Λ} and {Vβ : β ∈ Λ′} be two coverings of a space. {Uλ : λ ∈ Λ} is said to refine (or be a refinement of) {Vβ : β ∈ Λ′} if for each Uλ, there is some Vβ with Uλ ⊂ Vβ . Definition 4.2. Let {Uλ : λ ∈ Λ} be a covering of X. If Λ′ is contained in Λ and {Uλ : λ ∈ Λ′} is again a covering, it is called a subcovering. Definition 4.3. A covering {Uλ : λ ∈ Λ} of a topological space X is called locally finite if for every x ∈ X, there exists a neighborhood V of x such that Uλ ∩ V 6= ∅ only for a finite number of indexes.
