# Set theoretic and topological characterisations of ordered sets

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

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## 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 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.

Type of Work: Thesis (Doctorates > Ph.D.)
Award Type: Doctorates > Ph.D.
Supervisor(s):
Supervisor(s)EmailORCID
Good, ChristopherUNSPECIFIEDUNSPECIFIED
Licence:
College/Faculty: Colleges (2008 onwards) > College of Engineering & Physical Sciences
School or Department: School of Mathematics
Funders: None/not applicable
Subjects: Q Science > QA Mathematics
URI: http://etheses.bham.ac.uk/id/eprint/4388

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