On the degree spectrum of a $\Pi^0_1$ class

For any $\mathcal{P} \subseteq 2^{\omega}$, define $S(\mathcal{P})$, the degree spectrum of $\mathcal{P}$, to be the set of all Turing degrees $\bf {a}$ such that there exists $A\in \mathcal{P}$ of degree $\bf {a}$. We prove a number of basic properties of the structure which is the degree spectra of $\Pi^0_1$ classes ordered by inclusion and also study in detail some other phenomena relating to the study of $\Pi^0_1$ classes from a degree theoretic point of view, which are brought to light as a result of this analysis.