I am asked to prove or disprove that the Radial Plane is regular, completely regular and normal; using the following definition of its topology:
A set $A \subseteq \mathbb{R}^2$ is radially open if for every $x\in A$ we have that $A$ contains an open segment through $x$ in any direction.
If I am not mistaken I can prove that it is completely regular similarly to the Moore Plane (showing that for any open neighbourhood and a point $x$ in it I can found another neighbourhood whose closure is contained in the first one) using the segments through $x$, and because it is also $T_1$ then it would also be regular.
But I am struggling to find if it's normal or not. At first I thought about taking the sets $K = \{ (x,y) \ : \ x^2+y^2=1 \} \setminus \{(0,1)\}$ and $L = \{(0,1)\}$ but I am not that sure that $K$ is closed and even if it is I am not sure those two can't be separated by radially open sets. Any help? Thanks in advance