Reconstructing Topological Graphs and Continua

The deck of a topological space $X$ is the set $$\mathcal{D}$(X)={[X \setminus {x}] \colon x \in X}$,where$[Z]$denotesthehomeomorphismclassof$Z$. 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.