# Inverse systems with simplicial bonding maps and cell structures

## TL;DR

This paper constructs inverse systems of simplicial complexes with bonding maps that approximate a topologically complete space, establishing a homotopy equivalence and linking to cell structures.

## Contribution

It introduces a novel method to represent topologically complete spaces via inverse systems of simplicial complexes with specific bonding maps.

## Key findings

- Limit space is homotopy equivalent to the original space
- Provides a new connection between inverse systems and cell structures
- Extends understanding of topological space approximations

## Abstract

For a topologically complete space $X$ and a family of closed covers $\mathcal A$ of $X$ satisfying a "local refinement condition" and a "completeness condition," we give a construction of an inverse system $\mathbf{ N}_{\mathcal A}$ of simplicial complexes and simplicial bonding maps such that the limit space $N_{\infty} = \varprojlim \mathbf{N}_{\mathcal A}$ is homotopy equivalent to $X$. A connection with cell structures [2],[3] is discussed