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 $G_\delta$. 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 are presented, some reconstructible and others not reconstructible.Comment: 15 pages, 2 figure

Topologizing Group Actions

This thesis is centered on the following question: Given an abstract group action by an Abelian group G on a set X, when is there a compact Hausdorff topology on X such that the group action is continuous? If such a topology exists, we call the group action compact-realizable. We show that if G is a locally-compact group, a necessary condition for a G-action to be compact-realizable, is that the image of X under the stabilizer map must be a compact subspace of the collection of closed subgroups of G equipped with the co-compact topology. We apply this result to give a complete characterization for the case when G is a compact Abelian group in terms of the existence of continuous compact Hausdorff pre-images of a certain topological space associated with the group action. If G is not compact, we will show that the necessary condition is not sufficient. Together with various examples, we then present a general two-stage method of construction for compact Hausdorff topologies for ℝ-actions. For discrete groups, the necessary condition above turns out to be not very strong. In the case of G = ℤ2 we will see that the two cases |X| &lt; and |X| ≥ must be treated very differently. We derive necessary conditions for a group action with |X| &lt; to be compact-realizable by constructing particularly nice open partitions of the space X. We then use symbolic dynamics together with some generic constructions to obtain a partial converse in this case. If |X| ≥ we give further constructions of compact Hausdorff topologies for which the group action is continuous.</p

Topologizing group actions

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On affine groups admitting invariant two-point sets

A two-point set is a subset of the plane which meets every line in exactly two points. We discuss previous work on the topological symmetries of a two-point set, and show that there exist subgroups of S1 which do not leave any two-point set invariant. Further, we show that two-point sets may be chosen to be topological groups, in which case they are also homogeneous

Topological reconstruction and compactification theory

This thesis investigates the topological reconstruction problem, which is inspired by the reconstruction conjecture in graph theory. We ask how much information about a topological space can be recovered from the homeomorphism types of its point-complement subspaces. If the whole space can be recovered up to homeomorphism, it is called reconstructible. In the first part of this thesis, we investigate under which conditions compact spaces are reconstructible. It is shown that a non-reconstructible compact metrizable space must contain a dense collection of 1-point components. In particular, all metrizable continua are reconstructible. On the other hand, any first-countable compactification of countably many copies of the Cantor set is non-reconstructible, and so are all compact metrizable h-homogeneous spaces with a dense collection of 1-point components. We then investigate which non-compact locally compact spaces are reconstructible. Our main technical result is a framework for the reconstruction of spaces with a maximal finite compactification. We show that Euclidean spaces &reals;n and all ordinals are reconstructible. In the second part, we show that it is independent of ZFC whether the Stone-&Ccaron;ech remainder of the integers, &omega;&ast;, is reconstructible. Further, the property of being a normal space is consistently non-reconstructible. Under the Continuum Hypothesis, the compact Hausdorff space &omega;&ast; has a non-normal reconstruction, namely the space &omega;&ast;&bsol;&lcub;p&rcub; for a P-point p of &omega;&ast;. More generally, the existence of an uncountable cardinal &kappa; satisfying &kappa; = &kappa;&lt;&kappa; implies that there is a normal space with a non-normal reconstruction. The final chapter discusses the Stone-&Ccaron;ech compactification and the Stone-&Ccaron;ech remainder of spaces &omega;&ast;&bsol;&lcub;x&rcub;. Assuming the Continuum Hypothesis, we show that for every point x of &omega;&ast;, the Stone-&Ccaron;ech remainder of &omega;&ast;&lcub;x&rcub; is an &omega;2-Parovi&ccaron;enko space of cardinality 22c which admits a family of 2c disjoint open sets. This implies that under 2c = &omega;2, the Stone-&Ccaron;ech remainders of &omega;&ast;&bsol;&lcub;x&rcub; are all homeomorphic, regardless of which point x gets removed.This thesis is not currently available on ORA