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This is not a notion of continuity precisely in terms of preserving open sets, but perhaps one can make this connection by coming up with a suitably clever notion of an "open set" in a graph, e.g. involving closed edges. The property which I'm singling out as being suggestive of continuity is in this case the preservation of connectivity of subsets.

My question is: have people approached subjects such as tree-decompositions (or path decompositions, etc.) explicitly from a point of view of continuous functions before, and has it proven a useful way of presenting results? For instance, is there any reference which approaches the subject of tree-decompositions and tree-widths in this manner, or which discusses graph-theoretic structures in terms of such connectivity-preserving functions on vertex-sets?

(I posted this question previously on Math.SE, without any answers.)

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