We introduce radial variants of the Wijsman and Attouch-Wets topologies for the family $\mathcal{S}_{rc}^{d}$ of star-shaped sets (with respect to the origin) in $\mathbb{R}^{d}$ that are radially closed. These topologies give rise to new types of convergence for star-shaped sets, even when such sets are not closed or bounded. Our approach relies on a new family of functionals, called radial distance functionals, which measure “radial distances” between points and star-shaped sets in $\mathcal{S}_{rc}^{d}$ . These are natural radial analogues of the distance functionals for closed sets. However, unlike the radial functions of star-shaped sets, our radial distance functionals are real-valued maps and thus admit a natural treatment within the framework of classical function spaces. We prove that our radial Wijsman type topology $\tau _{W^{r}}$ is not metrizable on $\mathcal{S}_{rc}^{d}$ , while our radial Attouch-Wets type topology $\tau _{AW^{r}}$ is completely metrizable. A corresponding radial Attouch-Wets distance $d_{AW^{r}}$ is introduced, and we prove that $d_{AW}(A,K) \leq d_{AW^{r}}(A,K)$ for all closed $A,K \in \mathcal{S}_{rc}^{d}$ , where $d_{AW}$ denotes the Attouch-Wets distance.
