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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.

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I explore five topics in topology using set-theoretic techniques. The first of these is a generalization of 2-point sets called slice sets. I show that, for any small-in-cardinality subset A of the real line, there is a subset of the plane meeting every line in a topological copy of A. Under Martin's Axiom, I show how to improve this result to any totally disconnected A. Secondly, I show that it is consistent with and independent of ZFC to have a topologically rigid subset of the real line that is smaller than the continuum. Thirdly, I define and examine a new cardinal function related to cleavability. Fourthly, I explore the Toronto Problem and prove that any uncountable, Hausdorff, non-discrete Toronto space that is not regular falls into one of two strictly-defined classes. I also prove that for every infinite cardinality there are precisely 3 non-T1 Toronto spaces up to homeomorphism. Lastly, I examine a notion of dimension called the "neight", and prove several theorems that give a lower bound for this cardinal function.