Spaces where all closed sets are $\alpha$-limit sets

Abstract

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

Author Accepted Manuscript. Publication date: 1 April, 2022. Funded from the European Union's Horizon 2020 programme under the Marie Sklodowska-Curie Actions, project acronym "LISEDIDYS", grant agreement No. 883748

References in corpus (1)

• Folding points of unimodal inverse limit spaces