Use this URL to cite or link to this record in EThOS: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.581181
Title: On cleavability
Author: Levine, Shari
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2012
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Abstract:
This thesis concerns cleavability. A space X is said to be cleavable over a space Y along a set A subset of X if there exists a continuous function f from X to Y such that f(A) cap f(X setminus A) = emptyset. A space X is cleavable over a space Y if it is cleavable over Y along all subsets A of X. In this thesis we prove three results regarding cleavability. First we discover the conditions under which cleavability of an infinite compactum X over a first-countable scattered linearly ordered topological space (LOTS) Y implies embeddability of X into Y. In particular, we provide a class of counter-examples in which cleavability does not imply embeddability, and show that if X is an infinite compactum cleavable over ω1, the first uncountable ordinal, then X is embeddable into ω1. We secondly show that if X is an infinite compactum cleavable over any ordinal, then X must be homeomorphic to an ordinal. X must also therefore be a LOTS. This answers two fundamental questions in the area of cleavability. We also leave it as an open question whether cleavability of an infinite compactum X over an uncountable ordinal λ implies X is embeddable into λ. Lastly, we show that if X is an infinite compactum cleavable over a separable LOTS Y such that for some continuous function f from X to Y, the set of points on which f is not injective is scattered, then X is a LOTS. In addition to providing these three results, we introduce a new area of research developed from questions within cleavability. This area of research is called almost-injectivity. Given a compact T2 space X and a LOTS Y, we say a continuous function f from X to Y is almost-injective if the set of points on which f is not injective has countable cardinality. In this thesis, we state some questions concerning almost-injectivity, and show that if lambda is an ordinal, X is a T2 compactum, and f is an almost-injective function from X to lambda, then X must be a LOTS.
Supervisor: Knight, Robin; Peter, Collins Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID: uk.bl.ethos.581181  DOI: Not available
Keywords: Mathematics ; Analytic Topology or Topology ; cleavability ; compact ; continuous ; ordinal ; scattered ; homeomorphic
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