  In this paper it is proved that, when $Q$ is a quiver that admits some
closure, for any algebraically closed field $K$ and any finite dimensional
$K$-linear representation $\mathcal{X}$ of $Q$, if ${\rm
Ext}^1_{KQ}(\mathcal{X},KQ)=0$ then $\mathcal{X}$ is projective (Theorem 1.10).
In contrast, we show that if $Q$ is a specific quiver of the type above, then
there is an infinitely generated non-projective $KQ$-module $M_{\omega_1}$ such
that, when $K$ is a countable field, $\operatorname{\sf MA}_{\aleph_1}$
(Martin's Axiom for $\aleph_1$ many dense sets, which is a combinatorial axiom
in set theory) implies that ${\rm Ext}^1_{KQ}(M_{\omega_1},KQ)=0$ (Theorem
2.11).
