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

TL;DR

This paper investigates the reconstructibility of the space al* and normality, showing independence from ZFC axioms and that normality may not be reconstructible under certain set-theoretic assumptions.

Contribution

It proves that the reconstructibility of al* and the property of normality are independent of ZFC, highlighting set-theoretic conditions affecting topological reconstruction.

Findings

Reconstructibility of al* is independent of ZFC.

Under CH, al* can have a non-normal reconstruction.

Existence of certain uncountable cardinals implies non-normal reconstructibility.

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.