Fix an integer $n\ge 2$ and suppose that ${\cal L}$ is a set of [lines](https://en.wikipedia.org/wiki/Line_(geometry)#In_Euclidean_space) in $\mathbb{R}^n$. Is there a set $M\subseteq \mathbb{R}^n$ with the following properties?


1. $M$ intersects all the elements of ${\cal L}$, but
2. for all $m\in M$, the set $M\setminus\{m\}$ no longer intersects all the elements of $\cal L$.


In fact, I do not even know the answer for $n=2$.



Meeting a set of lines in $\mathbb{R}^n$
If we have a set $\mathcal L$ of lines in $\mathbb R^n$ such that $|\mathcal L|=\mathfrak c$, we can get the set $M$ with the desired properties using transfinite induction.


* Take any well-ordering of the set of lines $\mathcal L=\{l\_\alpha; \alpha<\mathfrak c\}$ (such that $\alpha\ne\beta$ implies $l\_\alpha\ne l\_\beta$.)
* By transfinite induction we define $M\_\alpha$ for $\alpha<\mathfrak c$ which is either singleton or empty set. If $l\_\alpha$ intersects $\bigcup\limits\_{\beta<\alpha} M\_\beta$, we put $M\_\alpha=\emptyset$. (We add nothing if $l\_\alpha$ already contains some points.) Otherwise we put $M\_\alpha=\{m\_\alpha\}$ where $m\_\alpha \in l\_\alpha \setminus \bigcup\limits\_{\beta<\alpha} l\_\beta$. (This set is non-empty, since $l\_\alpha$ has cardinality $\mathfrak c$ and the intersection $l\_\alpha \cap \bigcup\limits\_{\beta<\alpha} l\_\beta$ has cardinality smaller than $\mathfrak c$; each of the lines $l\_\beta$ can intersect $l\_\alpha$ at most in one point.)
* Now simply put $M=\bigcup\limits\_{\alpha<\mathfrak c} M\_\alpha$.


The set $M$ has the desired properties. Clearly, $M$ intersects each $l\_\alpha$. Moreover, if $m\in M$ and $m=m\_\alpha$, then $m$ is the only point on the line $l\_\alpha$. (If $m$ was added in the $\alpha$'s step, then $l\_\alpha$ does not contain any of the points from $\bigcup\limits\_{\beta<\alpha} M\_\beta$, i.e., the points added in the *preceding* steps. And the construction is done in such way that no new point on $l\_\alpha$ can be added in the steps *after* $\alpha$.) 




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If $|\mathcal L|<\mathfrak c$ then we can simply choose one point from each set $l \setminus \bigcup\limits\_{\substack{k\ne l\\ k\in \mathcal L}} k$. This set is non-empty - it has cardinality $\mathfrak c$. 




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I should point out that Gerhard Paseman suggested [in a comment](https://mathoverflow.net/questions/278143/meeting-a-set-of-lines-in-mathbbrn#comment685389_278143) another solution using transfinite induction. And for $n=2$ he suggested in [another comment](https://mathoverflow.net/questions/278143/meeting-a-set-of-lines-in-mathbbrn#comment685387_278143) a solution which does not use transfinite induction.




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Proof of the existence of Mazurkiewicz two-point set is in somewhat similar vein, see also this question: [Subset of the plane that intersects every line exactly twice](https://mathoverflow.net/q/21470).


