Papadopoulos, Kyriakos B.
(2013).
Set theoretic and topological characterisations of ordered sets.
University of Birmingham.
Ph.D.
Abstract
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 ordertheoretic 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.
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