Is there a subset of the real plane such that every line intersects that subset exactly once?

Question

Consider the real plane $\mathbb{R}^2$. Does there exist a subset $S$ of the real plane, such that every line $l$ in $\mathbb{R}^2$ intersects $S$ at exactly one point?

Answer 1

No.

Given $2$ distinct points $\mathbf{x}, \mathbf{y} \in S \subseteq \mathbb{R}^2$, consider the line $\ell$ that passes through $\mathbf{x}$ and $\mathbf{y}$. So $S$ can contain at most $1$ point. Obviously, $S \neq \varnothing$, so consider the cases where $|S| = 1$. It's easy to construct a line that misses the given point.