---
abstract: 'We provide an easily verifiable condition for local $k$-connectedness of an inverse limit of polyhedra.'
address:
- 'Department of Mathematics and Statistics, University of North Carolina at Greensboro, Greensboro, NC 27412, USA'
- 'Faculty of Mathematics, Informatics, and Mechanics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland'
author:
- 'G. C. Bell'
- 'A. Nagórko'
bibliography:
- 'references2.bib'
title: 'Local $k$-connectedness of an inverse limit of polyhedra'
---

[^1]

Introduction
============

The Nöbeling space characterization theorem [@ageev2007a; @ageev2007b; @ageev2007c; @levin2006; @nagorkophd] states that if a space is strongly universal in the class of $n$-dimensional separable complete metric spaces and is $k$-connected and locally $k$-connected for each $k < n$, then it is homeomorphic to the $n$-dimensional Nöbeling space $N^{2n+1}_n$.

In geometric group theory many spaces arise naturally as inverse limits of polyhedra (simplicial complexes endowed with the metric topology). In particular, boundaries at infinity of hyperbolic spaces can be expressed as such. Striking examples of applications of the Nöbeling space characterization theorem are proofs that the boundary at infinity of the curve complex of the $(n+5)$-punctured $2$-dimensional sphere is homeomorphic to the $n$-dimensional Nöbeling space $N^{2n+1}_n$ [@henselprzytycki2011; @gabai2014].

In the present paper we prove a condition for local 