On quasi-small loop groups
| Autor: | Behrooz Mashayekhy, Hanieh Mirebrahimi, Hamid Torabi, Ameneh Babaee |
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| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | Mathematica Slovaca. 72:1017-1030 |
| ISSN: | 1337-2211 0139-9918 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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