A Topological Variation of the Reconstruction Conjecture

This paper investigates topological reconstruction, related to the reconstruction conjecture in graph theory. We ask whether the homeomorphism types of subspaces of a space $X$ which are obtained by deleting singletons determine $X$ uniquely up to homeomorphism. If the question can be answered affirmatively, such a space is called reconstructible. We prove that in various cases topological properties can be reconstructed. As main result we find that familiar spaces such as the reals $\mathbb{R}$, the rationals $\mathbb{Q}$ and the irrationals $P$ are reconstructible, as well as spaces occurring as Stone-Cech compactifications. Moreover, some non-reconstructible spaces are discovered, amongst them the Cantor set $C$.