Orderability of spaces having ordered decompositions - Kemoto, Nobuyuki (Topology Proceedings, 2025)
The following may be well-known: - the subspace (0,1) {2} of the usual real numbers R is the topological sum of two linearly ordered spaces, and wellknown that there is no linear ordering of X whose open interval topology coincides with the topology of X. In this paper, we consider when the topological sum of a pairwise disjoint collection X of ordered spaces are orderable. As corollaries, we see: - whenever X contains infinitely many singletons or contains an infinite discrete space, its topological sum is orderable; - whenever X contains at least one ordered space with a maximal element but without minimal elements, its topological sum is orderable; - whenever X does not contain ordered spaces with both a maximal element and a minimal element, its topological sum is orderable; - whenever X contains infinitely many ordered spaces with both a maximal element and a minimal element, its topological sum is orderable; - whenever X consists of suborderable spaces, its topological sum is suborderable.