On representation spaces

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