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In this paper it is proved that, when $Q$ is a quiver that admits someclosure, for any algebraically closed field $K$ and any finite dimensional$K$-linear representation $\mathcal{X}$ of $Q$, if ${\rmExt}^1_{KQ}(\mathcal{X},KQ)=0$ then $\mathcal{X}$ is projective (Theorem 1.10).In contrast, we show that if $Q$ is a specific quiver of the type above, thenthere is an infinitely generated non-projective $KQ$-module $M_{\omega_1}$ suchthat, when $K$ is a countable field, $\operatorname{\sf MA}_{\aleph_1}$(Martin's Axiom for $\aleph_1$ many dense sets, which is a combinatorial axiomin set theory) implies that ${\rm Ext}^1_{KQ}(M_{\omega_1},KQ)=0$ (Theorem2.11).

Some infinitely generated non projective modules over path algebras and their extensions under Martin's Axiom