Topics in Computability

<p><strong>This thesis studies two topics in computability. The first is about computable metric and Polish spaces. We compare different notions of effective presentability and construct some spaces that are 'almost computable', in the sense that they do not have a computable presentation but they do have both left-c.e. and right-c.e. presentations. The second part studies c.e. Quasi-degrees (Q-degrees) and c.e. strong Quasi-degrees (sQ-degrees), which have interesting connections to algebra. We show that the c.e. sQ-degrees are not distributive, embed the lattice $N_5$ into them and show that no initial segment forms a lattice. We construct a non-computable c.e. set that has no c.e. simple set Q-below it. We also briefly study the relationship of sQ-degrees to wtt-degrees. Finally we show there is a minimal pair of sQ-degrees within the same Q-degree, and that if a degree is half of a minimal pair in the Q-degrees, it is also half of a minimal pair in the Turing degrees.</strong></p>