Is there set with no interior in $\mathbb{R}^2$, the intersection with $\lambda$-a.e. set $ \{x\} \times \mathbb{R}$ has an interior in $\mathbb{R}$? I am searching for a set that has no interior in $\mathbb{R}^2$ but when I intersect with the sets $\{x\} \times \mathbb{R}, x \in \mathbb{R}$ it has $\lambda$-a.e. an interior in $\mathbb{R}$, where $\lambda$ is the Lebesgue measure. The same should hold true for the sets $\mathbb{R} \times \{y\}, y \in \mathbb{R}$. My idea was to take $(q_n)_{n \in \mathbb{N}} = \mathbb{Q}^2$ and define $U = \bigcup_{n \in \mathbb{N}} B_{\frac{1}{2^n}}(q_n)$. Then I consider $A = U^C$, which has no interior in $\mathbb{R}^2$ since it contains no point of $\mathbb{Q}^2$. Now when I fix $x \in \mathbb{R}\setminus \mathbb{Q}$ does the set $C = A \cap (\{x\}\times \mathbb{R})$ have an interior?
 A: You could recursively construct a set that is dense in the plane, but intersects every line in at most two points. Its complement in the plane answers your question.
So called Mazurkiewicz set intersects every line in exactly two points, but to answer your question it is enough to construct a set dense in the plane that intersects every line in at most two points, which is easier to construct recursively in just countably many steps. Pick a countable base $\{U_n:n\ge 1\}$ for the topology of the plane, and at step $n$ pick a point $p_n\in U_n$ that does not lie on any of the (finitely many) lines that go through any two of the points that have already been picked at previous stages.
There is some literature about such sets, and I am bit lazy and only found one paper (because I wrote it) but you could find more along these lines (and, perhaps, better :). See
Non-regularity of some topologies on $\mathbb R^n$ stronger than the standard one.
Mathematica Pannonica (1994)
Volume: 5, Issue: 1, page 105-110.
https://eudml.org/doc/231430
In general this is also related to questions about joint continuity vs separate continuity, one could google that too for more papers.
Here, I remembered the term "two-point set", and googled some more links:
Can two-point sets be Borel?
https://mathoverflow.net/questions/272527/can-two-point-sets-be-borel
(there are more references if you follow the above link)
The axiom of choice and two-point sets in the plane
Arnold W. Miller
https://www.math.wisc.edu/~miller/res/two-pt.pdf
The Two Points Theorem of Mazurkiewicz,
Peter Komjath
Journal of Combinatorial Theory, Series A 99, 371–376 (2002)
doi:10.1006/jcta.2002.3278
https://www.sciencedirect.com/science/article/pii/S0097316502932784
