Let $n \geq 1$ be an integer. Let us call $S\subseteq \mathbb{R}^{n+1}$ an *$n$-sphere* if there is $x\in \mathbb{R}^{n+1}$ and $r\in \mathbb{R}$ with $r>0$ such that $$S = \{z\in \mathbb{R}^{n+1}: \|z-x\| = r\}, $$ where $\|\cdot\|$ denotes the Euclidean norm in $\mathbb{R}^{n+1}$.

Suppose that ${\cal S}\neq \emptyset$ is a set of $n$-spheres in $\mathbb{R}^{n+1}$.

Is there $M\subseteq \mathbb{R}^{n+1}$ such that $M$ intersects every member of ${\cal S}$, but for each $m\in M$ there is $S\in {\cal S}$ such that $S \cap (M\setminus \{m\}) = \emptyset$?

**Note**. The answer is positive if the set ${\cal S}$ is finite.