In this paper, we strengthen the properties of approximation by points (AP) and weak approximation by points (WAP) considered by A. Pultr and A. Tozzi in 1993 to define ${kappa}$-AP and ${kappa}$-WAP for an infinite cardinal ${kappa}$. We also strengthen the properties of radial and pseudoradial to define ${kappa}$-radial and ${kappa}$-pseudoradial for an infinite cardinal ${kappa}$. These allow us to consider new cardinal functions related to almost closed sets; AP-number, WAP-number, radial number, and pseudoradial number. We study their properties and show the relationships between them. We also provide some examples around ${kappa}$-AP and ${kappa}$-WAP which are closely connected with ${kappa}$-radial and ${kappa}$-pseudoradial.