Meeting a set of lines in $\mathbb{R}^n$

Question

Fix an integer $n\ge 2$ and suppose that ${\cal L}$ is a set of lines 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 $n=2$.

Answer 1

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 another solution using transfinite induction. And for $n=2$ he suggested in another comment 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.

Answer 2

For $n=2$ we define $M$ as follows:

$M$ is the union of the following sets:

1)The intersection with $x\_$ axis for lines not parallel to this axis.

2)The intersection with $y\_$axis for lines perpenedecular to this axis.