We say that a computable structure 
$\mathcal {A}$ is computably categorical if for every computable copy 
$\mathcal {B}$, there exists a computable isomorphism 
$f:\mathcal {A}\to \mathcal {B}$. This notion can be relativized to a degree 
$\mathbf {d}$ by saying that a computable structure 
$\mathcal {A}$ is computably categorical relative to 
$\mathbf {d}$ if for every 
$\mathbf {d}$-computable copy 
$\mathcal {B}$ of 
$\mathcal {A}$, there exists a 
$\mathbf {d}$-computable isomorphism 
$f:\mathcal {A}\to \mathcal {B}$. A key part of this thesis is to study the behavior of this notion of categoricity in the computably enumerable degrees.
The main theorem in Chapter 
$1$ states that given any computable partially ordered set P and any computable partition 
$P=P_0\sqcup P_1$, there exists an embedding h of P into the c.e. degrees and a computable graph 
$\mathcal {G}$ which is computably categorical, computably categorical relative to all degrees in 
$h(P_0)$, and is not computably categorical relative to any degree in 
$h(P_1)$. We also show that by using largely the same techniques alongside a standard construction of minimal pairs, we can embed a four-element diamond lattice into the c.e. degrees in the style of the main result of Chapter 
$1$.
We then apply some of the techniques used in this thesis to study the behavior of this notion in the context of generic degrees in Chapter 
$2$. Additionally, we show that several classes of structures admit a computable example that witnesses the pathological behavior of categoricity relative to a degree as seen in Chapter 
$1$’s main theorem.
Lastly, in the context of reverse mathematics, we investigate the reverse mathematical strength of a topological principle named 
$\mathsf {wGS}^{\operatorname {cl}}$, a weakened version of the Ginsburg–Sands theorem which states that every infinite topological space contains one of the following five topologies as a subspace, with 
$\mathbb {N}$ as the underlying set: discrete, indiscrete, cofinite, initial segment, or final segment.
Abstract prepared by Java Darleen Villano
E-mail: java.villano@utoronto.ca
URL: https://javavillano.github.io/thesisfinal.pdf
