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Let $G$ be locally finite graph. By $|G|$, we mean the Freudenthal compactification of the 1-complex $G$, see chapter 8 of "Graph Theory" by Diestel. It is followed from Lemma 8.5.4 of Diestel's book: Every connected standard subspace of $|G|$ is arc-connected. So if $|G|$ is a connected space, then $|G|$ is an arc-connected space, and so $G$ is connected as graph.

Now, my question is: Assume that $G$ is connected as graph. Is $|G|$ is a connected space?

It is true in the finite case. But I am not sure about the locally finite case.

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    $\begingroup$ What part of the proof for the finite case goes wrong in the locally finite case? I don't have the book to check the definitions, but since in a connected graph any two vertices are joined by a path of finite length, it seems to me that you should only have to look at finitely many vertices at a time. $\endgroup$ – Nate Eldredge Oct 12 '14 at 15:14
  • $\begingroup$ @NateEldredge infnite graphs have ends, so closed infinite subset of vertices may have end. $\endgroup$ – mojtaba Oct 12 '14 at 18:11
  • $\begingroup$ I'm afraid I still don't understand. Maybe I am naive, but the only definition of "connected graph" I know is that for every pair of vertices $a,b$, there is a finite list of vertices $x_0, x_1, \dots, x_n$ such that $x_0 = a$, $x_n = b$, and $x_i \sim x_{i+1}$. Is there a different definition which you are using? $\endgroup$ – Nate Eldredge Oct 12 '14 at 18:32
  • $\begingroup$ Any compactification of a connected space is connected. $\endgroup$ – Eric Wofsey Oct 12 '14 at 18:57
  • $\begingroup$ The closure of any connected set is connected. $\endgroup$ – Eric Wofsey Oct 14 '14 at 12:31

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