Sufficent Conditions for the preservation of Path-Connectedness in an arbitrary metric space

2024/04/24 by Andronicou, Savvas, Milakis, Emmanouil
#FOS: Mathematics #General Topology (math.GN)

paper · doi:10.48550/arxiv.2404.15871

Abstract

It is proven that if (X,d) is an arbitrary metric space and U is a path-connected subset of X with M:=\x_i: i∈\1,2,…,k\\⊂ int(U) , then the property of path-connectedness is also preserved in the resulting set U∖ M, provided that the boundary of each open ball of X is a non-empty and path-connected set. Moreover, under appropriate conditions we extend the above result in the case where the set M is countably infinite. As a consequence these results maintain path-connectedness for domains with holes.

Related

• Metric characterization of connectedness for topological spaces
• Metric and Path-Connectedness Properties of the Frechet Distance for Paths and Graphs
• Path-Connectedness of the Hyperspace of Compact Subsets of ℝⁿ
• Path Connectivity of Spheres in the Gromov-Hausdorff Class
• On punctured locally compact spaces