The lattice of convergence structures



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Many of the properties of the lattice of convergence structures on an arbitrary nonempty set may be exhibited in terms of the convergence structures under consideration. Lattices of several types of convergence structures of especial interest (e.g., the lattice of topologies) may be shown to be related to the lattice of convergence structures by means of particular lattice operators. Completions, in the lattice of convergence structures, of some subsets of special types of convergence structures may be given in terms of a family of convergence structures possessing a common property. In addition, the group of lattice automorphisms can be shown to be isomorphic to a clearly defined permutation group. By considering continuity of a self-map, two operators can be defined on the lattice of convergence structures. Each may be viewed in relationship to the types of convergence structures and/or lattice properties it preserves.