Thread: Spanning Trees for Infinite Graphs

Hi all,

I'm just a humble undergrad doing some research. I found that for one of my proofs, every step was equally valid for finite or infinite graphs, except one crucial argument.
At this step I used the fact that every graph has a spanning tree (if finite...) and proceeded to finish the proof.

Now I look back and wonder if can extend my result to the infinite case. However, as far as I can tell this would require a Spanning Tree Theorem for the infinite case. Anyone know anything about this?

Curious: Minimal spanning trees for weighted infinite graphs?
Curious: Introductory resources on infinite graphs?

Thanks in Advance

I suppose a definition for Infinite Tree would be required to answer this question. For my particular research topic, the following definition would be good for a tree:

Definition A tree is a graph for which there exist a unique path between any two vertices.

Note how this works for a vertex set of any cardinality (unless it doesn't, in which case I'm missing something and please feel free to point it out).

Now with this definition, does anyone know the answer to this question?