If $H= (V,E)$ is a hypergraph, a *matching* is a set $M\subseteq E$ such that $e_1\cap e_2 = \emptyset$ whenever $e_1\neq e_2 \in M$. The *matching number* $\mu(H)$ of a hypergraph $H=(V,E)$ with $V$ finite is the maximum number of elements a matching can have.

For infinite hypergraphs $H=(V,E)$, we let $$\mu(H) = \sup\{|M|:M\subseteq E\text{ is a matching}\}.$$ This definition agrees with the above definition for finite hypergraphs.

**Question.** If $H=(V,E)$ is a hypergraph with $V$ infinite, is there necessarily a matching $M_0\subseteq E$ such that $|M_0|=\mu(H)$?