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 \cite{higgs}, and graph-like spaces, which are topological spaces resembling graphs, introduced by Thomassen and Vella \cite{thomassenvella}. 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 show that an infinite graph has an abstract dual (which is a graph-like continuum) if and only if it is planar, and also that a graph-like continuum has an abstract dual (which is an infinite graph) if and only if it is planar. This generalizes theorems of Thomassen (\cite{thomassendual}) and Bruhn and Diestel (\cite{bruhndiestel}). The difficult part of the proof is extending Tutte's characterization of graphic matroids (\cite{tutte2}) to finitary or co-finitary B-matroids. In order to prove this characterization, we introduce a technique for obtaining these B-matroids as the limit of a sequence of finite minors. In \cite{tutte}, Tutte proved important theorems about the peripheral (induced and non-separating) circuits of a $3$-connected graph. He showed that for any two edges of a $3$-connected graph there is a peripheral circuit containing one but not the other, and that the peripheral circuits of a $3$-connected graph generate its cycle space. These theorems were generalized to $3$-connected binary matroids by Bixby and Cunningham (\cite{bixbycunningham}). We generalize both of these theorems to $3$-connected binary co-finitary B-matroids. Richter, Rooney and Thomassen \cite{richterrooneythomassen} showed that a locally connected, compact metric space has an embedding in the sphere unless it contains a subspace homeomorphic to $K_5$ or $K_{3,3}$, or one of a small number of other obstructions. We are able to extend this result to an arbitrary surface $\Sigma$; a locally connected, compact metric space embeds in $\Sigma$ unless it contains a subspace homeomorphic to a finite graph which does not embed in $\Sigma$, or one of a small number of other obstructions.

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 \cite{higgs}, and graph-like spaces, which are topological spaces resembling graphs, introduced by Thomassen and Vella \cite{thomassenvella}. 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 show that an infinite graph has an abstract dual (which is a graph-like continuum) if and only if it is planar, and also that a graph-like continuum has an abstract dual (which is an infinite graph) if and only if it is planar. This generalizes theorems of Thomassen (\cite{thomassendual}) and Bruhn and Diestel (\cite{bruhndiestel}). The difficult part of the proof is extending Tutte's characterization of graphic matroids (\cite{tutte2}) to finitary or co-finitary B-matroids. In order to prove this characterization, we introduce a technique for obtaining these B-matroids as the limit of a sequence of finite minors. In \cite{tutte}, Tutte proved important theorems about the peripheral (induced and non-separating) circuits of a $3$-connected graph. He showed that for any two edges of a $3$-connected graph there is a peripheral circuit containing one but not the other, and that the peripheral circuits of a $3$-connected graph generate its cycle space. These theorems were generalized to $3$-connected binary matroids by Bixby and Cunningham (\cite{bixbycunningham}). We generalize both of these theorems to $3$-connected binary co-finitary B-matroids. Richter, Rooney and Thomassen \cite{richterrooneythomassen} showed that a locally connected, compact metric space has an embedding in the sphere unless it contains a subspace homeomorphic to $K_5$ or $K_{3,3}$, or one of a small number of other obstructions. We are able to extend this result to an arbitrary surface $\Sigma$; a locally connected, compact metric space embeds in $\Sigma$ unless it contains a subspace homeomorphic to a finite graph which does not embed in $\Sigma$, or one of a small number of other obstructions.