Topological complexity of the telescope

TL;DR

This paper investigates the topological complexity of a mapping telescope, establishing an upper bound based on the complexities of the individual spaces in the sequence, providing new insights into their combined topological properties.

Contribution

It introduces an alternative definition of topological complexity and proves an upper bound for the complexity of the mapping telescope in terms of the maximum complexity of the constituent spaces.

Findings

Topological complexity of the telescope is bounded by twice the maximum complexity of the sequence spaces.

Provides a new approach to estimating complexity of complex topological constructions.

Enhances understanding of how topological complexity behaves under telescoping constructions.

Abstract

We use an alternative definition of topological complexity to show that the topological complexity of the mapping telescope of a sequence $X_1\rightarrow X_2\rightarrow X_3\rightarrow...$ is bounded above by $2max{TC(X_i); i=1,2,...}$.