Collections of paths and rays in the plane which fix its topology

A collection $\text{F}$ of proper maps into a locally compact Hausdorff space X fixes the topology of X if the only locally compact Hausdorff topology on X which makes each element of $\text{F}$ continuous and proper is the given topology. In I²=[-1, 1]×[-1, 1], neither the collection of analytic paths nor the collection of regular twice differentiable paths fixes the topology. However, in I², both the collection of C^∞ arcs and the collection of regular C¹ arcs fix the topology. In $\text{I}{}^2\text{=[?1,1]×[?1,1], the collection of polynomial rays together with any collection of paths does not fix the topology. However, in}\text{R}{}^2$, the collection of regular injective entire rays together with either the collection of C^∞ arcs or the collection of regular C¹ arcs fixes the topology.