I am reading the classical book Linear topological spaces by Kelley and Namioka. A set $A$ is called radial at $x$ if $A$ contains a segment starting at $x$ in any direction. An exercises states that there exists a subset of $\mathbb{R}^2$ which is radial at precisely one point.
It seems to me that this is related to the idea of radially interior point: the requested subset should have only one radially interior point. But I cannot figure out how I could construct such a set in the plane. I guess that it should have an empty interior (in the usual topology), but I need some help to go further.
I think there is an example in the book Real analysis and Probability, by Dudley et al.
In polar coordinates, the set $A$ is the union of all lines $\left\lbrace (r,\theta) \mid r \geq 0 \right\rbrace$ if $\theta / \pi$ is irrational, and of all the segments $$L_\theta = \left\lbrace (r,\theta) \mid 0 \leq r < \frac{1}{n} \right\rbrace$$ if $\theta / \pi = m/n$ is expressed in lowest terms. This set is radial at $0$ and at no other point.