## Cofinite graphs and their profinite completions

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We generalize the work of B. Hartley, to the category of topological graphs. The completion of a topological group in Hartley's work can be thought of as a particular example of the completion of a general uniform space. We consider topologies that are induced by uniformities, in order to topologize our graph structures and talk about their completions. In this dissertation generalize the idea of cofinite groups, due to B. Hartley. First we define cofinite spaces in general. Then, as a special case, we study cofinite graphs and their uniform completions. We are able to show that these completions are also cofinite graphs and being compact Hausdorff and totally disconnected they are rather regarded as profinite graphs. The idea of constructing a cofinite graph starts with defining a uniform topological graph in an appropriate fashion. We endow abstract graphs with uniformities corresponding to separating filter bases of equivalence relations with finitely many equivalence classes over the graph. By taking finitely many equivalence classes, we want to ensure the production of profinite structures over our topological graphs on taking the projective limit of the corresponding quotient graphs. It is established that for any cofinite graph there exists a unique cofinite completion. Generalizing Hartley's idea of cofinite groups and obtaining the structure for cofinite graphs we start establishing a parallel theory of cofinite graphs which in many ways can also be thought of as generalizations of the well-known works on pronite graphs by Pavel Zalesskii and Luis Ribes. Suitably defining the concept of cofinite connectedness of a cofinite graph we find that many of the properties of connectedness of topological spaces have analogs for cofinite connectedness. As an immediate consequence we obtain the following generalized characterization of the connected Cayley graphs of conite groups: G be a cofinite group, X be a cofinite space, then Cayley graph (G, X) is also cofinite graph and it is cofinitely connected if and only if X generates G (topologically). Our immediate next concern is developing group actions on cofinite graphs. Defining the action of an abstract group over a cofinite graph in the most natural way we are able to characterize a unique way of uniformizing an abstract group with a cofinite structure, obtained from the cofinite structure of the graph in the underlying action, so that the afore said action becomes uniformly continuous. We show that the aforesaid actions can actually be extended to the structures' of corresponding cofinite completions, preserving the underlying character of the original group action.