Convergence in Topological Spaces

In this thesis the writer has considered types of convergence in an arbitrary topological space. Three types of convergence are considered, convergence of sequences in the real numbers with the usual topology, convergence of Moore-Smith sequences in an arbitrary topological space, and convergence of filters in an arbitrary topological space.In Chapter II, we show that sequences are inadequate to describe limit points of sets and hence the topology in an arbitrary topological space. The idea of a sequence is generalized in this chapter to Moore-Smith sequences and in Chapter III to filters.In Chapter II we prove that convergence of Moore-Smith sequences is sufficient to describe limit points, closed sets, the closure of a set, open sets, and in fact the topology of an arbitrary space. Although the convergence of filters and Moore-Smith sequences differ greatly, we prove these same results using filters. However, in Chapter II we also prove an iterated limit theorem using generalized Cauchy sequences in a complete metric space.Chapter IV is in many respects the most important, although containing the fewest results. For in Chapter IV we are able to relate convergence of filters to convergence of Moore-Smith sequences. In fact we prove that filters and Moore-Smith sequences are equivalent convergence theories.