Reconstructing topological graphs and continua
Volume 148 / 2017
Colloquium Mathematicum 148 (2017), 107-122
MSC: Primary 05C60, 54E45; Secondary 54B05, 54D05, 54D35, 54F15.
DOI: 10.4064/cm7011-10-2016
Published online: 24 February 2017
Abstract
The deck of a topological space $X$ is the set ${\mathcal D}(X) =\{[X \setminus \{x\}] : x \in X\}$, 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.
