Reconstructing compact metrizable spaces
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- by Paul Gartside, Max F. Pitz and Rolf Suabedissen PDF
- Proc. Amer. Math. Soc. 145 (2017), 429-443 Request permission
Abstract:
The deck, $\mathcal {D}(X)$, of a topological space $X$ is the set $\mathcal {D}(X) =\{[X \setminus \{x\}]:x \in X\}$, where $[Y]$ denotes the homeomorphism class of $Y$. A space $X$ is (topologically) reconstructible if whenever $\mathcal {D}(Z)=\mathcal D(X)$, then $Z$ is homeomorphic to $X$. It is known that every (metrizable) continuum is reconstructible, whereas the Cantor set is non-reconstructible.
The main result of this paper characterises the non-reconstructible compact metrizable spaces as precisely those where for each point $x$ there is a sequence $\langle B_n^x:n \in \mathbb {N}\rangle$ of pairwise disjoint clopen subsets converging to $x$ such that $B_n^x$ and $B_n^y$ are homeomorphic for each $n$ and all $x$ and $y$.
In a non-reconstructible compact metrizable space the set of $1$-point components forms a dense $G_\delta$. For $h$-homogeneous spaces, this condition is sufficient for non-reconstruction. A wide variety of spaces with a dense $G_\delta$ set of $1$-point components is presented, some reconstructible and others not reconstructible.
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Additional Information
- Paul Gartside
- Affiliation: The Dietrich School of Arts and Sciences, 301 Thackeray Hall, Pittsburgh, Pennsylvania 15260
- Email: gartside@math.pitt.edu
- Max F. Pitz
- Affiliation: Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
- Address at time of publication: Department of Mathematics, University of Hamburg, Bundesstraße 55 (Geomatikum), 20146 Hamburg, Germany
- Email: max.pitz@uni-hamburg.de
- Rolf Suabedissen
- Affiliation: Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
- Email: rolf.suabedissen@maths.ox.ac.uk
- Received by editor(s): October 19, 2015
- Received by editor(s) in revised form: March 10, 2016
- Published electronically: July 22, 2016
- Additional Notes: The second author is the corresponding author
This research formed part of the second author’s thesis at the University of Oxford (2015), supported by an EPSRC studentship. - Communicated by: Mirna Džamonja
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 429-443
- MSC (2010): Primary 54E45; Secondary 05C60, 54B05, 54D35
- DOI: https://doi.org/10.1090/proc/13270
- MathSciNet review: 3565393