  In this paper, we study some properties of homotopical closeness for paths.
We define the quasi-small loop group as the subgroup of all classes of loops
that are homotopically close to null-homotopic loops, denoted by $\pi_1^{qs}
(X, x)$ for a pointed space $(X, x)$. Then we prove that, unlike the small loop
group, the quasi-small loop group $\pi_1^{qs}(X, x)$ does not depend on the
base point, and that it is a normal subgroup containing $\pi_1^{sg}(X, x)$, the
small generated subgroup of the fundamental group. Also, we show that a space
$X$ is homotopically path Hausdorff if and only if $\pi_1^{qs} (X, x)$ is
trivial. Finally, as consequences, we give some relationships between the
quasi-small loop group and the quasi-topological fundamental group.
