  Given a topological Ramsey space $(\mathcal R,\leq, r)$, we extend the notion
of semiselective coideal to sets $\mathcal H\subseteq\mathcal R$ and study
conditions for $\mathcal H$ that will enable us to make the structure
$(\mathcal R,\mathcal H,\leq, r)$ a Ramsey space (not necessarily topological)
and also study forcing notions related to $\mathcal H$ which will satisfy
abstract versions of interesting properties of the corresponding forcing
notions in the realm of Ellentuck's space. This extends results of Farah, and
results of Mijares, to the most general context of topological Ramsey spaces.
As applications, we prove that for every topological Ramsey space $\mathcal R$,
under suitable large cardinal hypotheses every semiselective ultrafilter
$\mathcal U\subseteq\mathcal R$ is generic over $L(\mathbb R)$; and that given
a semiselective coideal $\mathcal H\subseteq\mathcal R$, every definable subset
of $\mathcal R$ is $\mathcal H$--Ramsey. This generalizes the corresponding
results for the case when $\mathcal R$ is equal to Ellentuck's space.
