Let $G = (V, E)$ be a simple, undirected graph. We consider the following two definitions of graph connectedness:


(1) $G$ is connected if for $x,y \in V$ there is a finite path connecting $x$ and $y$.


(2) If $A \subseteq V$ such that $A \neq \emptyset$ and $A \neq V$ there is $e\in E$ such that $e \cap A \neq \emptyset$ and $e \cap (V\setminus A) \neq \emptyset$.


It is not hard to see that for finite graphs, (1) and (2) are equivalent, and that (1) implies (2) in the general case.


Is there an infinite graph that is connected with in the sense of definition (2), but not in the sense of definition (1)?



Two definitions of graph connectedness
To be more explicit, fix a vertex a and let A0 be the singleton set having a as a member. Using the mechanism of (2) and some version of the axiom of choice, define An+1 by adding the one vertex that is guaranteed to be adjacent to but not in the vertex set An. Take the union of these sets and call it A. Now either A is all of V or else something went horribly, horribly wrong. The remaining details of (1) are left to the (horrified) reader.


Gerhard "And I Do Mean Horribly" Paseman, 2012.05.14


