Topological Coarse Shape Homotopy Groups

• Published in: arXiv.org
• Main Authors: Ghanei, Fateme, Mirebrahimi, Hanieh, Mashayekhy, Behrooz, Nasri, Tayyebe
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 04.04.2016
• Subjects: Topology
• ISSN: 2331-8422

Uchillo-Ibanez et al. introduced a topology on the sets of shape morphisms between arbitrary topological spaces in 1999. In this paper, applying a similar idea, we introduce a topology on the set of coarse shape morphisms \(Sh^*(X,Y)\), for arbitrary topological spaces \(X\) and \(Y\). In particular, we can consider a topology on the coarse shape homotopy group of a topological space \((X,x)\), \(Sh^*((S^k,*),(X,x))=\check{\pi}_k^{*}(X,x)\), which makes it a Hausdorff topological group. Moreover, we study some properties of these topological coarse shape homotopoy groups such as second countability, movability and in particullar, we prove that \(\check{\pi}_k^{*^{top}}\) preserves finite product of compact Hausdorff spaces. Also, we show that for a pointed topological space \((X,x)\), \(\check{\pi}_k^{top}(X,x)\) can be embedded in \(\check{\pi}_k^{*^{top}}(X,x)\).