  The deck of a topological space $X$ is the set $\mathcal{D}(X)=\{[X \setminus
\{x\}] \colon x \in X\}$, where $[Z]$ denotes the homeomorphism class of $Z$. A
space $X$ is topologically reconstructible if whenever
$\mathcal{D}(X)=\mathcal{D}(Y)$ then $X$ is homeomorphic to $Y$.
  It is shown that all metrizable compact connected spaces are reconstructible.
It follows that all finite graphs, when viewed as a 1-dimensional cell-complex,
are reconstructible in the topological sense, and more generally, that all
compact graph-like spaces are reconstructible.
