The space $\omega^*$ and the reconstruction of normality

Abstract

The topological reconstruction problem asks how much information about a topological space can be recovered from its point-complement subspaces. If the whole space can be recovered in this way, it is called reconstructible. Our main result states that it is independent of the axioms of set theory (ZFC) whether the Stone-Cech remainder of the integers $\omega^*$ is reconstructible. Our second result is about the reconstruction of normality. We show that assuming the Continuum Hypothesis, the compact Hausdorff space $\omega^*$ has a non-normal reconstruction, namely the space $\omega^* \setminus \{p\}$ for a $P$-point $p$ of $\omega^*$. More generally, we show that the existence of an uncountable cardinal $\kappa$ satisfying $\kappa = \kappa^{<\kappa}$ implies that there is a normal space with a non-normal reconstruction. These results demonstrate that consistently, the property of being a normal space is not reconstructible. Whether normality is non-reconstructible in ZFC remains an open question.