arXiv subjects

Rolf Suabedissen

Publications and source records attributed to Rolf Suabedissen.

5 recordsLinked to original sources

arXiv2015

Reconstructing Compact Metrizable Spaces

The deck, $\mathcal{D}(X)$, of a topological space $X$ is the set $\mathcal{D}(X)=\{[X \setminus \{x\}]\colon x \in X\}$, 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 $\langle B_n^x \colon n \in \mathbb{N}\rangle$ 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 $G_δ$. For $h$-homogeneous spaces, this condition is sufficient for non-reconstruction. A wide variety of spaces with a dense $G_δ$ set of $1$-point components are presented, some reconstructible and others not reconstructible.

math.GN↗

arXiv2015

Reconstructing Topological Graphs and Continua

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.

math.GN↗

arXiv2014

The Stone-Cech compactifications of $ω^*\setminus \{x\}$ and $S_κ\setminus\{x\}$

The space $S_κ$ is the Stone space of the $κ$-saturated Boolean algebra of cardinality $κ$. It exists provided that $κ= κ^{<κ}$, and is characterised topologically as the unique $κ$-Parovichenko space of weight $κ$. Under the Continuum Hypothesis, $S_{ω_1}$ coincides with $ω^*$. This paper investigates questions related to the Stone-Cech compactification of spaces $S_κ\setminus \{x\}$, extending corresponding results obtained by Fine & Gillman and Comfort & Negrepontis for the space $ω^*$. We show that for every point $x$ of $S_κ$, the Stone-Cech remainder of $S_κ\setminus \{x\}$ is a $κ^+$-Parovichenko space of cardinality $2^{2^κ}$ which admits a family of $2^κ$ disjoint clopen sets. As a corollary we get that it is consistent with CH that the Stone-Cech remainders of $ω^* \setminus \{x\}$ are all homeomorphic.

math.GN↗

arXiv2013

A Topological Variation of the Reconstruction Conjecture

This paper investigates topological reconstruction, related to the reconstruction conjecture in graph theory. We ask whether the homeomorphism types of subspaces of a space $X$ which are obtained by deleting singletons determine $X$ uniquely up to homeomorphism. If the question can be answered affirmatively, such a space is called reconstructible. We prove that in various cases topological properties can be reconstructed. As main result we find that familiar spaces such as the reals $\mathbb{R}$, the rationals $\mathbb{Q}$ and the irrationals $P$ are reconstructible, as well as spaces occurring as Stone-Cech compactifications. Moreover, some non-reconstructible spaces are discovered, amongst them the Cantor set $C$.

math.GN↗

arXiv2013

Finite compactifications of $ω^* \setminus \{x\}$

We prove that under [CH], finite compactifications of $ω^* \setminus \{x\}$ are homeomorphic to $ω^*$. Moreover, in each case, the remainder consists almost exclusively of $P$-points, apart from possibly one point. Similar results are obtained for other, related classes of spaces, amongst them $S_κ$, the $κ$-Parovičenko space of weight $κ$. Also, some parallels are drawn to the Cantor set and the Double Arrow space.

math.GN↗