I think this is actually a pretty interesting question, so I'm not sure why it has been voted down. Let me briefly expand on Qiaochu's comment. Suppose I ask the related question:
Does the notion of an infinite cycle in an infinite graph G make sense?
Essentially, the answer is no in the graph G itself. However, it turns out that if we add extra points to G by suitably compactifying then the answer is yes. This is the approach taken by Diestel and Bruhn. It turns out that the cycles we obtain after compactifying are well-behaved in the sense that they still satisfy some basic properties that finite cycles in finite graphs do.
A different approach is to start with the notion of cycle itself and ask which kinds of objects have a reasonable notion of cycles. It turns out that graph-like spaces are a rich source of such objects. Indeed, this line of work can be viewed as a generalization of the Diestel and Bruhn approach, since the Freudenthal compactification of a locally finite infinite graph is a graph-like space. Finite graphs are also graph-like spaces, and there are graph-like spaces which do not arise by compactifying infinite graphs.