Examples of Closure Operators
Most of the examples presented in this chapter will be used throughout the book, especially the closure operators for topological spaces, R-modules and for groups presented in sections 3.3, 3.4, and 3.5, respectively. Nevertheless, we begin with structures which generalize topological spaces, namely pretopological spaces and filter convergence spaces, for two reasons. First, additive and grounded closure operators of concrete categories may be interpreted as concrete functors with values in the category of pretopological spaces, as we shall see in Chapter 5 and apply in Chapter 8. Second, the natural closure operators of these generalized topological structures are closely linked to the natural closure operators occurring in the categories of graphs and partially ordered sets presented in 3.6. Hence they provide a unifying view of topological and “discrete” structures.
KeywordsTopological Space Maximal Subgroup Closure Operator Bottom Element Topological Category
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- The category FC contains the cartesian closed topological category of all Choquet spaces or pseudotopological spaces as a full subcategory, and PrTop is still fully contained in that subcategory: see Bentley and Lowen  for details. We refer also to Bentley, Herrlich and Lowen-Colebunders  for further reading on generalized convergence structures. Chapters 6–8 contain an abundance of closure operators of Top which make the four given examples of 3.3 look like chosen rather arbitrarily; however, these four are most frequently used as building blocks for new operators. Moreover, the Kuratowski closure operator can be characterized as the only non-trivial hereditary and additive closure operator of the subcategory of T0-spaces with “good behaviour” on products (called finite structure property, see 4.11); this and further characterizations are given in Dikranjan, Tholen and Watson . Readers interested in (pre)radicals of modules and Abelian groups should consult Fuchs  and Bican, J amb or, Kepka and Nemec . Finally, we note that domain theory is still a fast-growing branch in the interface between mathematics and computer science. Almost any of the many expository articles and texts will inspire the reader to establish new closure operators in the categories in question.Google Scholar