The Finite Coarse Shape Paths

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(pn), 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