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

TL;DR

This paper investigates metrizable spaces where every closed set is an $ ext{alpha}$-limit set for some continuous map, showing that such spaces include those with many arcs but are not preserved under common topological operations.

Contribution

It characterizes spaces with enough arcs where all closed sets are $ ext{alpha}$-limit sets and analyzes how this property behaves under various topological constructions.

Findings

Spaces with enough arcs have all closed sets as $ ext{alpha}$-limit sets.

The property is not preserved under sums, products, or quotients.

Constructed spaces from spaces with enough arcs retain the property.

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.