Reconstructing compact metrizable spaces (Q2832841)

Reconstructing topological spaces from their (abstract) groups of homeomorphisms is well-studied in the literature. The present paper deals with a different approach to reconstructing topological spaces that finds its origin in graph theory. The authors define a space \(X\) to be \textit{reconstructible} if whenever \(\mathcal{D}(Z)=\mathcal{D}(X)\), then \(Z\) is homeomorphic to \(X\). Here \(\mathcal{D}(X) = \{[X\setminus \{x\}]: x\in X\}\), where \([Y]\) denotes the homeomorphism class of \(Y\). It is known that every metrizable continuum is reconstructible, whereas the familiar Cantor set is not. A usable, nontrivial and interesting topological characterization of the non-reconstructible compact metrizable spaces is obtained. They are precisely the compact metrizable spaces \(X\) where for each point \(x\) in \(X\) there is a sequence \(\{B_x^n : n \in \mathbb{N}\}\) of pairwise disjoint nonempty clopen subsets of \(X\) converging to \(x\) such that \(B_x^n\) and and \(B_y^n\) are homeomorphic for each \(n\) and all \(x\) and \(y\). Every non-reconstructible compact metrizable space contains a dense \(G_\delta\)-set of 1-point components. The converse of this is not true. Several interesting examples are also constructed.