  For a ring R, Hilbert's Tenth Problem HTP(R) is the set of polynomial
equations over R, in several variables, with solutions in R. We consider
computability of this set for subrings R of the rationals. Applying Baire
category theory to these subrings, which naturally form a topological space,
relates their sets HTP(R) to the set HTP($\mathbb{Q}$), whose decidability
remains an open question. The main result is that, for an arbitrary set C,
HTP($\mathbb{Q}$) computes C if and only if the subrings R for which HTP(R)
computes C form a nonmeager class. Similar results hold for 1-reducibility, for
admitting a Diophantine model of $\mathbb{Z}$, and for existential definability
of $\mathbb{Z}$.
