On the Computer-Checked Solution of the Kuratowski Closure-Complement Problem
نویسنده
چکیده
Since its beginnings back in 1989, the development of the Mizar Mathematical Library is strongly stimulated by the formalization of general topology. In the paper we describe a solution of the famous Kuratowski closure-complement (and, as a side-effect, the interiorcomplement) problem, which are more a kind of mathematical puzzle (or exercise for students) than regular mathematics. Based on the large Mizar Mathematical Library, especially using its well-developed part devoted to the topology, we construct also examples in the topology of a real line illustrating both exercises. We were surprised by the influence of our solution of this problem on the improvement of the whole Mizar library.
منابع مشابه
On the Kuratowski Closure-Complement Problem
In this article we formalize the Kuratowski closure-complement result: there is at most 14 distinct sets that one can produce from a given subset A of a topological space T by applying closure and complement operators and that all 14 can be obtained from a suitable subset of R, namely KuratExSet = {1} ∪Q(2, 3) ∪ (3, 4) ∪ (4,∞). The second part of the article deals with the maximal number of dis...
متن کاملThe Kuratowski Closure-complement Theorem
This remarkable little theorem and related phenomena have been the concern of many authors. Apart from the mysterious appearance of the number 14, the attraction of this theorem is that it is simple to state and can be examined and proved using concepts available after any first encounter with topology. The goal of this article is both to provide an original investigation into variations of the...
متن کاملSome (Fuzzy) Topologies on General\ Fuzzy Automata
In this paper, by presenting some notions and theorems, we obtaindifferent types of fuzzy topologies. In fact, we obtain someLowen-type and Chang-type fuzzy topologies on general fuzzyautomata. To this end, first we define a Kuratowski fuzzy interioroperator which induces a Lowen-type fuzzy topology on the set ofstates of a max- min general fuzzy automaton. Also by provingsome theorems, ...
متن کاملClosures in Formal Languages and Kuratowski's Theorem
A famous theorem of Kuratowski states that in a topological space, at most 14 distinct sets can be produced by repeatedly applying the operations of closure and complement to a given set. We re-examine this theorem in the setting of formal languages, where closure is either Kleene closure or positive closure. We classify languages according to the structure of the algebra they generate under it...
متن کاملAn Approach for Solving Traveling Salesman Problem
In this paper, we introduce a new approach for solving the traveling salesman problems (TSP) and provide a solution algorithm for a variant of this problem. The concept of the proposed method is based on the Hungarian algorithm, which has been used to solve an assignment problem for reaching an optimal solution. We introduced a new fittest criterion for crossing over such problems, and illu...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2004