On cleavability

Thesis

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.

Authors: Levine, S

• Publication date: 2012
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: ordinal; scattered; compact; cleavability; LOTS; homeomorphic; continuous
• Subjects: Mathematics; Analytic Topology or Topology