  For a (not necessarily locally convex) topological vector space $\mathcal{X}$
of holomorphic functions in one complex variable, we show that the shift
invariant subspace generated by a set of polynomials is $\mathcal{X}$ if and
only if their common vanishing set contains no point at which the evaluation
functional is continuous. For two variables, we show that this problem can be
reduced to determining the cyclicity of a single polynomial and obtain partial
results for more than two variables. We proceed to examine the maximal domain,
i.e., the set of all points for which the evaluation functional is continuous.
When $\mathcal{X}$ is metrizable, we show that the maximal domain must be an
$F_\sigma$ set, and then construct Hilbert function spaces on the unit disk
whose maximal domain is the disk plus an arbitrary subset of the boundary that
is both $F_\sigma$ and $G_\delta$.
