Affiliation:
1. Department of Pure Mathematics Center of Excellence in Analysis on Algebraic Structures Ferdowsi University of Mashhad P.O.Box 1159-91775 Mashhad Iran
Abstract
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.