In this talk we will present some new results about cardinal inequalities on topological spaces. We introduce the cardinal invariant $nu_s(X)$, the non-Urysohn number for singletons, to generalize the Urysohn separation axiom. Using this invariant, we generalize and extend some known cardinal inequalities for Urysohn spaces to all topological spaces, particularly such that involve variations of tightness and pseudocharcter. The main results pertain to upper bounds on the cardinalities of closures and $\theta$-closures of sets, and variations of the Arhangelskii-Sapirovskii inequality.