Topological reconstruction and compactification theory

Thesis

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 ℝⁿ 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, ω&ast;, is reconstructible. Further, the property of being a normal space is consistently non-reconstructible. Under the Continuum Hypothesis, the compact Hausdorff space ω&ast; has a non-normal reconstruction, namely the space ω&ast;&bsol;&lcub;p&rcub; for a P-point p of ω&ast;. More generally, the existence of an uncountable cardinal κ satisfying κ = κ^<κ 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 ω&ast;&bsol;&lcub;x&rcub;. Assuming the Continuum Hypothesis, we show that for every point x of ω&ast;, the Stone-&Ccaron;ech remainder of ω&ast;&lcub;x&rcub; is an ω₂-Parovi&ccaron;enko space of cardinality 2^2c which admits a family of 2^c disjoint open sets. This implies that under 2^c = ω₂, the Stone-&Ccaron;ech remainders of ω&ast;&bsol;&lcub;x&rcub; are all homeomorphic, regardless of which point x gets removed.

Authors: Pitz, M; Max Pitz

• Publication date: 2015
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: Oxford University, UK
• Language: English
• Keywords: Topological Reconstruction; Reconstruction Conjecture
• Subjects: Mathematics; Analytic Topology or Topology