Topology and Infinite Graphs

The main focus of this paper will be on two very different areas in which topology is relevant to the study of infinite graphs. The first is the mechanics of compactness proofs, which use a particular group of lemmas to extend results about finite subgraphs to apply to an entire infinite graph. We will explore these results by using them to prove a result of de Bruijn and Erdos, that an infinite graph is k-colorable if its finite subgraphs are k-colorable, in several different ways. The second area is a relatively new area of study pioneered by Diestel which redefines certain concepts of graph theory in terms of a topology on a graph. Specifically, we find that certain basic features of the cycle space cannot be extended verbatim to infinite graphs. But if we define the cycle space in terms of homeomorphic images of the circle S1 in a compactified topology on the graph, we can find extensions. This will be motivated in more detail and some of the consequences explored