On Representation Spaces

Let C be a class of topological spaces, let P be a subset of C, and let α be a class of mappings having the composition property. Given X ∈ C, we write X ∈ Clα(P) if for every open cover U of X there is a space Y ∈ P and a U-mapping ƒ: X → Y that belongs to α. The closure operator Clα defines a topology τα in C. After proving general properties of the operator Clα, we investigate some properties of the topological space (ℕ, τα), where ℕ is the space of all nondegenerate metric continua and α is one of the following classes: all mappings, confluent mappings, or monotone mappings