Reconstructing topological graphs and continua

Paul Gartside ¹   ; Max F. Pitz ²   ; Rolf Suabedissen ³

¹ The Dietrich School of Arts and Sciences 301 Thackeray Hall Pittsburgh, PA 15260, U.S.A.
² Department of Mathematics University of Hamburg Bundesstraße 55 (Geomatikum) 20146 Hamburg, Germany
³ Mathematical Institute University of Oxford Oxford OX2 6GG, United Kingdom

Colloquium Mathematicum, Tome 148 (2017) no. 1, pp. 107-122

Résumé

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.

Détail

DOI : 10.4064/cm7011-10-2016

Keywords: deck topological space set mathcal setminus where denotes homeomorphism class space emph topologically reconstructible whenever mathcal mathcal homeomorphic shown metrizable compact connected spaces reconstructible follows finite graphs viewed dimensional cell complex reconstructible topological sense generally compact graph like spaces reconstructible