The Finite Coarse Shape Paths

Ivan Jelić () and Ivančica Mirošević
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Ivan Jelić: Faculty of Science, University of Split, 21000 Split, Croatia
Ivančica Mirošević: Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, 21000 Split, Croatia

Mathematics, 2025, vol. 13, issue 3, 1-13

Abstract: In this paper, we introduce the notions of finite coarse shape path and finite coarse shape path connectedness of a topological space. We prove that the solenoid Σ ( p n ) , which is known to be coarse shape path connected but not shape path connected, is not finite coarse shape path connected either. Furthermore, we show that every finite coarse shape path induces an isomorphism between finite coarse shape groups of the topological space at different base points, with some interesting and useful properties. We also show that finite coarse shape groups of the same space, in general, depend on the choice of a base point. Hence, the pointed finite coarse shape type of X , x , in general, depends on the choice of the point x . Finally, we prove that if X is a finite coarse shape path connected paracompact locally compact space, then the pointed finite coarse shape type of X , x does not depend on the choice of the point x .

Keywords: topological space; inverse system; pro ? -category; shape; coarse shape; finite coarse shape; polyhedron; connectedness; finite coarse shape path; solenoid (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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