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Variations on Kuratowski’s 14-set theorem. (English) Zbl 1210.54004

Summary: Kuratowski’s 14-set theorem says that in a topological space, 14 is the maximum possible number of distinct sets which can be generated from a fixed set by taking closures and complements. Here we consider the analogous question for any possible subcollection of the operations \(\{\)closure, complement, interior, intersection, union\(\}\), and any number of initially given sets. Even though these problems concern point-set topology, the solutions rely instead on lattice theory and related flavors of universal algebra, with a little bit of logic for seasoning.

MSC:

54A05 Topological spaces and generalizations (closure spaces, etc.)
05A15 Exact enumeration problems, generating functions
06B99 Lattices
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