On $\Sigma_1$-Definable Closed Unbounded Sets

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.