Title:

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 pointcomplement 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 nonreconstructible compact metrizable space must contain a dense collection of 1point components. In particular, all metrizable continua are reconstructible. On the other hand, any firstcountable compactification of countably many copies of the Cantor set is nonreconstructible, and so are all compact metrizable hhomogeneous spaces with a dense collection of 1point components. We then investigate which noncompact 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 ℝ^{n} and all ordinals are reconstructible. In the second part, we show that it is independent of ZFC whether the StoneČech remainder of the integers, ω*, is reconstructible. Further, the property of being a normal space is consistently nonreconstructible. Under the Continuum Hypothesis, the compact Hausdorff space ω* has a nonnormal reconstruction, namely the space ω*\{p} for a Ppoint p of ω*. More generally, the existence of an uncountable cardinal κ satisfying κ = κ^{<κ} implies that there is a normal space with a nonnormal reconstruction. The final chapter discusses the StoneČech compactification and the StoneČech remainder of spaces ω*\{x}. Assuming the Continuum Hypothesis, we show that for every point x of ω*, the StoneČech remainder of ω*{x} is an ω_{2}Parovičenko space of cardinality 2^{2c} which admits a family of 2^{c} disjoint open sets. This implies that under 2^{c} = ω_{2}, the StoneČech remainders of ω*\{x} are all homeomorphic, regardless of which point x gets removed.
