Topological spaces induced by homotopic distance
الموضوعات :
T. Vergili
1
(
Department of Mathematics, Faculty of Science, Karadeniz Technical University, Trabzon, Turkey
)
A. Borat
2
(
Department of Mathematics, Faculty of Engineering and Natural Sciences, Bursa Technical University, Bursa, Turkey
)
الکلمات المفتاحية: Metric spaces, Topological complexity, Homotopic distance, Lusternik Schnirelmann category,
ملخص المقالة :
Topological complexity which plays an important role in motion planning problem can be generalized to homotopic distance $\mathrm{D}$ as introduced in \cite{MVML}. In this paper, we study the homotopic distance and mention that it can be realized as a pseudometric on $\mathrm{Map}(X,Y)$. Moreover we study the topology induced by the pseudometric $\mathrm{D}$. In particular, we consider the space $\mathrm{Map}(S^1,S^1)$ and use the non-compactness of it to talk about the non-compactness of $\mathrm{Map}(X,Y)$.
[1] P. Bubenik, T. Vergili, Topological Spaces of Persistence Modules and their properties, J. Appl. Comput. Topol. 2 (2018), 233-269.
[2] O. Cornea, G. Lupton, J. Oprea, D. Tanre, Lusternik-Schnirelmann category, Mathematical Surveys and Monographs, 103, American Mathematical Society, 2003.
[3] E. Cech, Topological Spaces, John Wiley & Sons, 1966.
[4] M. Farber, Topological complexity of motion planning, Discrete and Computational Geometry. 29 (2003), 211-221.
[5] J. C. Latombe, A. Lazanas, S. Shekhar, Robot motion planning with uncertainty in control and sensing, Artificial Intelligence. 52 (1991), 1-47.
[6] E. Macıas-Virgós, D. Mosquera-Lois, Homotopic distance between maps, Math. Proc. Cambridge Philos. Soc. 172 (1) (2021), 73-93.
[7] J. Oprea, J. Strom, Mixing categories, Proc. Amer. Math. Soc. 139 (9) (2011), 3383-3392.
[8] A. Rieser, Cech closure spaces: A unified framework for discrete and continuous homotopy, Topol. Appl. (2021), 296:107613.
[9] E. H. Spanier, Algebraic Topology, McGraw-Hill, 1966.
