Use this URL to cite or link to this record in EThOS: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.575646
Title: Set theoretic and topological characterisations of ordered sets
Van Dalen and Wattel have shown that a space is LOTS (linearly orderable topological space) if and only if it has a T$$_1$$-separating subbase consisting of two interlocking nests. Given a collection of subsets $$\mathcal L$$ of a set X, van Dalen and Wattel define an order $$\triangleleft$$$$_\mathcal L$$ by declaring $$_\mathcal X$$ $$\triangleleft$$$$_\mathcal L$$ $$_\mathcal Y$$ if and only if there exists some L $$\in$$ $$\mathcal L$$ such that x $$\in$$ L but y $$\notin$$ L. We examine $$\triangleleft$$$$_\mathcal L$$ in the light of van Dalen and Wattel’s theorem. We go on to give a topological characterisation of ordinal spaces, including $$_\mathcal W$$$$_1$$, in these terms, by first observing that the T$$_1$$-separating union of more than two nests generates spaces that are not of high order-theoretic interest. In particular, we give an example of a countable space X, with three nests $$\mathcal L$$,$$\mathcal R$$,$$\mathcal P$$, each T$$_0$$-separating X, such that their union T$$_1$$-separates X, but does not T$$_2$$-separate X. We then characterise ordinals in purely topological terms, using neighbourhood assignments, with no mention of nest or of order. We finally introduce a conjecture on the characterisation of ordinals via selections, which may lead into a new external characterisation.