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## Every vertex belongs to some component even in infinite graphs? [closed]

Defining a component as a maximally connected subgraph, is there a way to prove (or a counter-example) that every vertex belongs to some component, even in the case of infinite graphs? I know you can just build the components for finite graphs, but how would you ensure maximality in the infinite case?

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The component of a vertex $v$ consists of all the vertices that can be connected to $v$ by paths in the graph. I'm voting to close, as this is not a research-level question. – Andreas Blass Jul 18 2011 at 16:21