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 ∈ Cl α ( P ) if for every open cover U of X there is a space Y ∈ P and a U -mapping f : X → Y that belongs to α. The closure operator Cl α defines a topology τ α in C . After proving general properties of the operator Cl α , we investigate some properties of the topological space ( N , τ α ) , 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.