  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 $C$.
