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 ω<sub>1</sub>, the first uncountable ordinal, then X is embeddable into ω<sub>1</sub>. 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 T<sub>2</sub> 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 T<sub>2</sub> compactum, and f is an almost-injective function from X to lambda, then X must be a LOTS.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:581181 |
| Date | January 2012 |
| Creators | Levine, Shari |
| Contributors | Knight, Robin; Peter, Collins |
| Publisher | University of Oxford |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://ora.ox.ac.uk/objects/uuid:feb28947-c21a-4149-b1d4-e21548da8af5 |
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