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.



Meeting a set of spheres in $\mathbb{R}^n$
Let $(S\_i)\_{i \in I}$ be a family of spheres of cardinality $|I| < \mathfrak{c}$ and let $S$ be a sphere distinct from each $S\_i$. Each intersection $S \cap S\_i$ is a (possibly degenerate) circle. Since there is a continuum of circles inside $S$, one can find a circle $C \subseteq S$ distinct from each $S \cap S\_i$.
Each intersection $C \cap S \cap S\_i$ consists of at most two points. Since $|I| < \mathfrak{c} = |C|$, the family $(C \cap S \cap S\_i)\_{i \in I}$ does not cover $C$. In particular $S \setminus \bigcup\_{i \in I} S\_i$ is non-empty.


Using this and applying the method used by Martin Sleziak to answer your previous question [Meeting a set of lines in $\mathbb{R}^n$](https://mathoverflow.net/questions/278143/meeting-a-set-of-lines-in-mathbbrn) (or as in Will Brian's answer above), one gets a positive answer in ZFC.


Remark: it is annoying that you do not link to your previous very similar questions ([here](https://mathoverflow.net/questions/278143/meeting-a-set-of-lines-in-mathbbrn), [here](https://mathoverflow.net/questions/280014/meeting-a-set-of-lines-in-a-generalization-of-mathbbrn), [here](https://mathoverflow.net/questions/278584/minimal-subcoverings-of-a-cover-with-finite-intersection), and also [here](https://mathoverflow.net/questions/278586/minimal-subcoverings-of-a-cover-whose-sets-intersect-in-at-most-1-point)).


