From: Julian Dontchev Newsgroups: sci.math.research Subject: Re: How many sets are generated by closure, union and complement Date: 14 Sep 1998 10:10:14 GMT David McIntyre wrote: : Given a finite family F of subsets of a topological space X, is there a : finite family G with F \subset G which is closed under the operations of : closure, union and complement? : It is a well-known result of Kuratowski (Fund Math 3, 1922) that closure : and complement applied to a single set yield at most 14 distinct subsets : of X. I seem to recall that the number of sets which can be generated : from a single set using closure, union and complement is similarly : bounded, although I can't easily find a reference to this. Hello Dave, check one of those 3 papers, pehaps you find nothing but who knows... --------------- (1) Fife, James H, The Kuratowski closure-complement problem, Math. Mag. 64, No.3, 180-182 (1991). [ISSN 0025-570X] The author presents an easily readable proof of the famous Kuratowski closure-complement problem, similar to, but not identical with Kuratowski's original [{\it C. Kuratowski}, Sur l'op eration A de l'analysis situs, Fundam. Math. 3, 182-199 (1922)]. (2) Moslehian, M.S.; Tavallaii, N., A generalization of the Kuratowski closure-complement problem, J. Math., Punjab Univ. 28, 1-9 (1995). By considering the interval $(0,1)$ on the real line it is easy to show that it is not possible, in general, to obtain the boundary of a given set using complementation, closure and interior operations on that set. Therefore one can generalize the Kuratowski closure-complement problem in a special sense. In this paper, we show that if any pair of operations among closure, interior, boundary and complementations may be chosen, then using only two of the operations on any given set $X$ we obtain that $X$ belongs to a specific finite family. In addition, for any pair of these operations we give necessary and sufficient conditions that the related family possesses largest cardinal number. (3) Fife, James H., The Kuratowski closure-complement problem, Math. Mag. 64, No.3, 180-182 (1991). [ISSN 0025-570X] The author presents an easily readable proof of the famous Kuratowski closure-complement problem, similar to, but not identical with Kuratowski's original [{\it C. Kuratowski}, Sur l'op eration A de l'analysis situs, Fundam. Math. 3, 182-199 (1922)]. ============================================================================== From: Fred Galvin Newsgroups: sci.math.research Subject: Re: How many sets are generated by closure, union and complement Date: Tue, 15 Sep 1998 01:15:59 -0500 On Mon, 14 Sep 1998, David McIntyre wrote: [same quotation as above, deleted -- djr] I seem to recall that you can generate an infinite number of sets that way. For some problems of this kind which do have finite bounds, see the American Mathematical Monthly, April, 1988, p. 353; e.g., using complement, interior, and boundary, you get a maximum of 34 sets (Edwin Buchman, problem E3144); also see the references given in the editorial comment.