MacLane's Theorem for Graph-Like Spaces
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The cycle space of a finite graph is the subspace of the edge space generated by the edge sets of cycles, and is a well-studied object in graph theory. Recently progress has been made towards extending the theory of cycle spaces to infinite graphs. Graph-like spaces are a class of topological objects that reconcile the combinatorial properties of infinite graphs with the topological properties of finite graphs. They were first introduced by Thomassen and Vella as a natural, general class of topological spaces for which Menger's Theorem holds. Graph-like spaces are the natural objects for extending classical results from topological graph theory and cycle space theory to infinite graphs. This thesis focuses on the topological properties of embeddings of graph-like spaces, as well as the algebraic properties of graph-like spaces. We develop a theory of embeddings of graph-like spaces in surfaces. We also show how the theory of edge spaces developed by Vella and Richter applies to graph-like spaces. We combine the topological and algebraic properties of embeddings of graph-like spaces in order to prove an extension of MacLane's Theorem. We also extend Thomassen's version of Kuratowski's Theorem for 2-connected compact locally connected metric spaces to the class of graph-like spaces.