Cardinality bounds involving the skew-$\lambda$ Lindel\"of degree and its variants

We introduce a modified closing-off argument that results in several improved bounds for the cardinalities of Hausdorff and Urysohn spaces. These bounds involve the cardinal invariant $skL(X,\lambda)$, the skew-$\lambda$ Lindel\"of degree of a space $X$, where $\lambda$ is a cardinal. $skL(X,\lambda)$ is a weakening of the Lindel\"of degree and is defined as the least cardinal $\kappa$ such that if $\mathcal{U}$ is an open cover of $X$ then there exists $\mathcal{V}\in [\mathcal{U}]^{\leq\kappa}$ such that $|X\backslash\cup\mathcal{V}|<\lambda$. We show that if $X$ is Hausdorff then $|X|\leq 2^{skL(X,\lambda)t(X)\psi(X)}$, where $\lambda= 2^{t(X)\psi(X)}$. This improves the well-known Arhangel'skii- \v{S}apirovskii bound $2^{L(X)t(X)\psi(X)}$ for the cardinality of a Hausdorff space $X$. We additionally define several variations of $skL(X,\lambda)$, establish other related cardinality bounds, and provide examples.