  The deck, $\mathcal{D}(X)$, of a topological space $X$ is the set
$\mathcal{D}(X)=\{[X \setminus \{x\}]\colon x \in X\}$, where $[Y]$ denotes the
homeomorphism class of $Y$. A space $X$ is (topologically) reconstructible if
whenever $\mathcal{D}(Z)=\mathcal{D}(X)$ 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 $\langle B_n^x \colon n \in \mathbb{N}\rangle$ of pairwise disjoint
clopen subsets converging to $x$ such that $B_n^x$ and $B_n^y$ 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 $G_\delta$. For $h$-homogeneous spaces, this condition
is sufficient for non-reconstruction. A wide variety of spaces with a dense
$G_\delta$ set of $1$-point components are presented, some reconstructible and
others not reconstructible.
