  Definable stationary sets, and specifically, ordinal definable ones, play a
significant role in the study of canonical inner models of set theory and the
class HOD of hereditarily ordinal definable sets. Fixing a certain notion of
definability and an uncountable cardinal, one can consider the associated
family of definable closed unbounded sets. In this paper, we study the extent
to which such families can approximate the full closed unbounded filter, and
their dependence on the defining complexity. Focusing on closed unbounded
subsets of a cardinal $\kappa$ which are $\Sigma_1$-definable in parameters
from $H_\kappa$ and ordinal parameters, we show that the ability of such closed
unbounded sets to well approximate the closed unbounded filter on $\kappa$ can
highly vary, and strongly depends on key properties of the underlying universe
of set theory.
