Reconstructing compact metrizable spaces

The deck, D(X), of a topological space X is the set D(X) = {[X\{x}]: x ∈ X}, where [Y] denotes the homeomorphism class of Y. A space X is (topologically) reconstructible if whenever D(Z) = D(X), then Z is homeomorphic to X. It is known that every (metrizable) continuum is reconstructible, whereas the Cantor set is non-reconstructible. The main result of this paper characterises the non-reconstructible compact metrizable spaces as precisely those where for each point x there is a sequence <B^x/_n: n ∈ ℕ> of pairwise disjoint clopen subsets converging to x such that B^x_n and B^y_n are homeomorphic for each n and all x and y. In a non-reconstructible compact metrizable space the set of 1-point components forms a dense G_δ. For h-homogeneous spaces, this condition is sufficient for non-reconstruction. A wide variety of spaces with a dense G_δ set of 1-point components is presented, some reconstructible and others not reconstructible.