Traditional graph theory focuses on finite graphs. Two vertices are considered connected iff there is a finite walk between them (basically a sequence of vertices, each one adjacent to the last).
If we start considering infinite graphs, we have to consider some interesting consequences. Suppose we want to consider vertices x,y as connected by an infinite walk. Then we can order this walk linearly, and get a sequence $x=x_1,x_2,x_3,\ldots,x_\infty=y$. If this is an infinite sequence, it must contain some limit point -- some point $x_k$ must have either no immediate predecessor or no immediate successor. This idea doesn't match the old definition of a walk between vertices.
Although I don't know of any references, I do think this idea is interesting as a way to extend connected concepts to infinite graphs.
For example, consider the tree of surreal numbers. Think of the surreals as sign sequences; each is a map from an ordinal to $\{-,+\}$. Say that $s$ and $t$ are adjacent if the domain of $s$ is one more or one less than the domain of $t$. So 0 is adjacent to -1 and 1, 1 is adjacent to 0, 2, and 1/2, etc.
This looks like a standard tree until we get surreals with infinite domains. Consider some $s$ with domain $\omega$, and the sequence $s_i$ where $s_i$ has domain $i = \{j:j, and $s_i(j)=s(j)$. It feels like the $s_i$ walk should connect 0 with $s$. A new definition to capture this idea would be interesting.