A cover $\mathcal C$ of a set $X$ by subsets of $X$ is called

$\bullet$ *minimal* if for every $C\in\mathcal C$ the family $\mathcal C\setminus\{C\}$ is not a cover of $X$;

$\bullet$ *minimizable* if $\mathcal C$ contains a minimal subcover of $X$.

For example, any cover of the plane by parallel lines is minimal.

Problem.Is each cover of the plane $\mathbb R^2$ by lines minimizable? What is the answer for the rational plane $\mathbb Q^2$?

**Acknowledgement.** The problem was motivated by this question of Dominic van der Zypen.

**Added in Edit.** For the rational plane the affirmative answer can be also deduced from the following general

Theorem.A countable hypergraph $H=(V,\mathcal E)$ admits a minimal set $\mathcal M\subset \mathcal E$ with $\bigcup \mathcal M=\bigcup \mathcal E$ if each infinite set of vertices $I\subset V$ contains a finite set $F\subset I$ such that the set of edges $\{E\in \mathcal E:F\subset E\}$ is finite.

The proof of this theorem is a bit complicated and will be presented in our joint paper with Dominic van der Zypen.

We do not know if this theorem holds for arbitrary (not necessarily countable) hypergraphs.

On the other hand, @Peter Komjath in his comment to the answer of @bof claims that he can prove that every cover of the real plane by lines is minimizable, using some (difficult) general result on minimal covers of hypergraphs whose edges have the same cardinality and have small intersections.