On quasi-small loop groups

Abstract

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 π 1 q s ( X , x ) $\pi_1^{qs} (X, x)$ for a pointed space (X, x). Then we prove that, unlike the small loop group, the quasi-small loop group π 1 q s ( X , x ) $\pi_1^{qs}(X, x)$ does not depend on the base point, and that it is a normal subgroup containing π 1 s g ( X , x ) $\pi_1^{sg}(X, x)$ , the small generated subgroup of the fundamental group. Also, we show that a space X is homotopically path Hausdorff if and only if π 1 q s ( X , x ) $\pi_1^{qs} (X, x)$ is trivial. Finally, as consequences, we give some relationships between the quasi-small loop group and the quasi-topological fundamental group.