We study closed subspaces of $\kappa$-Ohio complete spaces and, for $\kappa$ uncountable cardinal, we prove a characterization for them. We then investigate the behaviour of products of $\kappa$-Ohio complete spaces. We prove that, if the cardinal ${\kappa }^{+}$ is endowed with either the order or the discrete topology, the space ${\left({\kappa }^{+}\right)}^{{\kappa }^{+}}$ is not $\kappa$-Ohio complete. As a consequence, we show that, if $\kappa$ is less than the first weakly inaccessible cardinal, then neither the space ${\omega }^{{\kappa }^{+}}$, nor the space ${ℝ}^{{\kappa }^{+}}$ is $\kappa$-Ohio complete.