Infinite graphs, graph-like spaces and B-matroids

I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. iii The central theme of this thesis is to prove results about infinite mathematical objects by studying the behaviour of their finite substructures. In particular, we study B-matroids, which are an infinite generalization of matroids introduced by Higgs [13], and graph-like spaces, which are topological spaces resembling graphs, introduced by Thomassen and Vella [27]. Recall that the circuit matroid of a finite graph is a matroid defined on the edges of the graph, with a set of edges being independent if it contains no circuit. It turns out that graph-like continua and infinite graphs both have circuit B-matroids. The first main result of this thesis is a generalization of Whitney’s Theorem that a graph has an abstract dual if and only if it is planar. We sho