Orderability of spaces having ordered decompositions
Keywords:
orderable, suborderable, topological sum, ordered setAbstract
The following may be well-known:
- the subspace $(0,1) \cup\{2\}$ of the usual real numbers $\mathbb{R}$ 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 $\mathcal{X}$ of ordered spaces are orderable. As corollaries, we see:
- whenever $\mathcal{X}$ contains infinitely many singletons or contains an infinite discrete space, its topological sum is orderable;
- whenever $\mathcal{X}$ contains at least one ordered space with a maximal element but without minimal elements, its topological sum is orderable;
- whenever $\mathcal{X}$ does not contain ordered spaces with both a maximal element and a minimal element, its topological sum is orderable;
- whenever $\mathcal{X}$ contains infinitely many ordered spaces with both a maximal element and a minimal element, its topological sum is orderable;
- whenever $\mathcal{X}$ consists of suborderable spaces, its topological sum is suborderable.
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