Reconstructing Topological Graphs and Continua

TL;DR

This paper proves that all metrizable compact connected spaces, including finite graphs and graph-like spaces, can be uniquely reconstructed from their topological deck, establishing a significant class of reconstructible spaces.

Contribution

It demonstrates that all metrizable compact connected spaces are topologically reconstructible, extending to finite graphs and graph-like spaces.

Findings

All metrizable compact connected spaces are reconstructible.

Finite graphs are reconstructible as 1-dimensional cell complexes.

Compact graph-like spaces are also reconstructible.

Abstract

The deck of a topological space $X$ is the set $\mathcal{D}(X)=\{[X \setminus \{x\}] \colon x \in X\}$, where $[Z]$ denotes the homeomorphism class of $Z$. A space $X$ is topologically reconstructible if whenever $\mathcal{D}(X)=\mathcal{D}(Y)$ then $X$ is homeomorphic to $Y$. It is shown that all metrizable compact connected spaces are reconstructible. It follows that all finite graphs, when viewed as a 1-dimensional cell-complex, are reconstructible in the topological sense, and more generally, that all compact graph-like spaces are reconstructible.