Spaces where all closed sets are α-limit sets

Metrizable spaces are studied in which every closed set is an α-limit for some continuous map and point. It shown that this property enjoyed by space containing sufficiently many arcs (formalized the notion of a with enough arcs), though such need not be arcwise connected. Further it preserved topological sums, products images quotients. However, positive results do hold metrizable obtained those constructions from arcs.