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Kuratowski Theorems in Soft Topology

Gültekin SOYLU, Müge ÇERÇİ
2020 Adıyaman üniversitesi fen bilimleri dergisi  
First the closure operator is investigated in the soft topological setting and afterwards the Kuratowski Closure-Complement Theorem is stated and proved.  ...  This paper deals with the soft topological counterparts of concepts introduced by Kuratowski.  ...  The correspondence of a closure operator with a soft topology is given. In addition the well-known Kuratowski Closure-Complement Theorem is stated and proved for soft topological spaces.  ... 
doi:10.37094/adyujsci.597721 fatcat:36gbqfkmrjfqzdwqib3f7hwfdu

Generalization of Kuratowski problem in linear spaces [article]

Allahkaram Shafie
2019 arXiv   pre-print
The highest number of distinct sets that can be generated from one convex set in linear space by repeatedly applying algebraic closure and complement in any order is 8.  ...  Keywords: Kuratowski problem, algebraic interior, algebraic closure  ...  The notations algebraic closure,complement and algebraic interior will be denoted, respectively by f, g, h. According to Lemma (1.1) (b) we have f hA = f A, hf A = hA which implies that f f A = f A.  ... 
arXiv:1902.03299v1 fatcat:5jqb7x746zaytpdzuzj5y6h7h4

A NOTE ON ΑIG- CLOSURE AND ΑIG- INTERIOR IN IDEAL TOPOLOGICAL SPACES

D. Vinodhini, Et. al.
2021 Turkish Journal of Computer and Mathematics Education  
Also, it is established that αIg- closure is a Kuratowski closure operator on (X, t, I) under certain conditions.  ...  The concepts of αIg- closure, αIg- interior and αIg- boundary of a subset of an ideal topological space (X, t, I) are introduced in this article. Some of their basic properties are proven.  ...  Kuratowski's closure complement theorem [1] has been a guiding source of topology.  ... 
doi:10.17762/turcomat.v12i4.1189 fatcat:pzlp67odgfhadarxlalfed6bdu

A Kuratowski closure-complement variant whose solution is independent of ZF [article]

Michael P. Cohen, Todd Johnson, Adam Kral, Aaron Li, Justin Soll
2020 arXiv   pre-print
We pose the following new variant of the Kuratowski closure-complement problem: How many distinct sets may be obtained by starting with a set A of a Polish space X, and applying only closure, complementation  ...  The set operator d was studied by Kuratowski in his foundational text Topology: Volume I; it assigns to A the collection dA of all points of second category for A.  ...  (Contrast with the situation in the original closure-complement problem, and in Theorem 2.)  ... 
arXiv:2005.13009v1 fatcat:fbpkgqafpfbk7hcyldnth2xzi4

Kuratowski's Theorem for Two Closure Operators [article]

Jeffrey Shallit, Ross Willard
2011 arXiv   pre-print
A celebrated 1922 theorem of Kuratowski states that there are at most 14 distinct sets arising from applying the operations of complementation and closure, any number of times, in any order, to a subset  ...  In this paper we consider the case of complementation and two abstract closure operators.  ...  Let Cl(A) denote the topological closure of A, and c(A) denote S − A, the complement of A.  ... 
arXiv:1109.1227v1 fatcat:c5hqbbbr5beobjtdbde73yt2jm

The radical-annihilator monoid of a ring

Ryan C. Schwiebert
2016 Communications in Algebra  
Kuratowski's closure-complement problem gives rise to a monoid generated by the closure and complement operations.  ...  We succeed in characterizing semiprime rings and commutative dual rings by their radical-annihilator monoids, and we determine the monoids for commutative local zero-dimensional (in the sense of Krull  ...  Acknowledgments The author wishes to thank the referee for his careful reading and numerous helpful recommendations.  ... 
doi:10.1080/00927872.2016.1222401 fatcat:fipzvjie4bgephl2f6ay26fubu

Kuratowski Monoids of n-Topological Spaces

Taras Banakh, Ostap Chervak, Tetyana Martynyuk, Maksym Pylypovych, Alex Ravsky, Markiyan Simkiv
2018 Topological Algebra and its Applications  
Generalizing the famous 14-set closure-complement Theorem of Kuratowski from 1922, we prove that for a set X endowed with n pairwise comparable topologies Ʈ1 ⊂ · · · ⊂ Ʈn, by repeated application of the  ...  operations of complement and closure in the topologies Ʈ1, . . . , Ʈn to a subset A ⊂ X we can obtain at most  ...  Introduction This paper was motivated by the famous Kuratowski 14-set closure-complement Theorem [4] , which says that the repeated application of the operations of closure and complement to a subset  ... 
doi:10.1515/taa-2018-0001 fatcat:dgbmo24trvfsxnlx2oqe4dooq4

Variations on Kuratowski's 14-set theorem [article]

David Sherman
2004 arXiv   pre-print
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.  ...  In this article we consider the analogous questions for any possible subcollection of the operations closure, complement, interior, intersection, union, and any number of initially given sets.  ...  The author has benefited from several conversations with John D'Angelo.  ... 
arXiv:math/0405401v1 fatcat:g4hc5fds4jfqnfcq4mvwzqab24

Page 420 of American Mathematical Society. Bulletin of the American Mathematical Society Vol. 45, Issue 6 [page]

1939 American Mathematical Society. Bulletin of the American Mathematical Society  
Dense: The interior of its closure is the space; the boundary of its complement is the closure of its complement; its complement is a boundary set; its closure is a non-boundary set. II.  ...  A non-boundary set: The closure of its interior is the s pace; the boundary of the closure of its complement ts the closure of its complement; its complement is nondense; its interior is dense.  ... 

Kuratowski monoids of n-topological spaces [article]

T. Banakh, O. Chervak, T. Martynyuk, M. Pylypovych, A. Ravsky, M. Simkiv
2015 arXiv   pre-print
Generalizing the famous 14-set closure-complement Theorem of Kuratowski from 1922, we prove that for a set X endowed with n pairwise comparable topologies τ_1⊂...  ...  ⊂τ_n, by repeated application of the operations of complement and closure in the topologies τ_1,...,τ_n to a subset A⊂ X we can obtains at most 2K(n)=2∑_i,j=0^ni+jii+jj distinct sets.  ...  Introduction This paper was motivated by the famous Kuratowski 14-set closure-complement Theorem [1] , which says that the repeated application of the operations of closure and complement to a subset  ... 
arXiv:1508.07703v4 fatcat:clqz52dz3bcxfam6d7frrdomue

E -Local Functions And Ue -Operator In Ideal Topological Spaces

G. Saravanakumar, S. Tamilselvan, A. Vadivel
2017 Zenodo  
The main goal of this paper is to introduce another local function to give possibility of obtaining a Kuratowski closure operator.  ...  -local functions for the ideal topological spaces has been described within this work. Moreover, with the help of -local functions Kuratowski closure operators and topology are obtained.  ...  But in [8, 14] and [15] we are not able to define a Kuratowski Closure operator with the help of () local function.  ... 
doi:10.5281/zenodo.845915 fatcat:2jmsxz3xjjhgtl2rbf27sptbne

On non-boundary sets

A. D. Wallace
1939 Bulletin of the American Mathematical Society  
A set is dense if its closure is the space; boundary if its complement is dense; nondense if its closure is boundary; and finally, non-boundary if its complement is nondense.  ...  We adhere to the nomenclature of Kuratowski's Topologie] except as noted. In particular, we suppose that 5 is a non vacuous space satisfying his axioms of closure, and we write i ?  ...  Dense: The interior of its closure is the space; the boundary of its complement is the closure of its complement; its complement is a boundary set; its closure is a non-boundary set. II.  ... 
doi:10.1090/s0002-9904-1939-06997-8 fatcat:qd2g3nfcojeffheshrjvp5chb4

Page 17 of American Mathematical Society. Bulletin of the American Mathematical Society Vol. 30, Issue [page]

1924 American Mathematical Society. Bulletin of the American Mathematical Society  
Kuratowski, Casimir. On the accessibility of an arc from its complement in space of three dimensions, 31, 32. —— See Knaster, B. Kuratowski, Casimir, and Zarankiewicz, Casimir.  ...  Three theorems on closure of biorthogonal systems of functions, 33, 97. — On the zeros of exponential sums and integrals, 37, 213. —— An inverse problem in differential equations, 39, 814. —— The asymptotic  ... 

Page 5081 of Mathematical Reviews Vol. , Issue 81M [page]

1981 Mathematical Reviews  
Theorem 1: If (X,C) is a complemented cocompact closure algebra, then (My,7) is compact.  ...  Theorem 2: Every compact Hausdorff space with a basis consisting of clopen sets is homeomorphic to the dual space of a complemented cocompact closure algebra. J. Kardasek (Brno) 81m:54008  ... 

Pythagorean Nano Topological Space

2020 International journal of recent technology and engineering  
Our main objective is to introduce the concept of Pythagorean nano fuzzy topological space by motivating from the notion of fuzzy topological space and nano topological space and properties of Pythagorean  ...  Theorem 3.10: (Kuratowski closure axiom) The Pythagorean nano closure in a PNT space is the Kuratowski closure operator.  ...  Thus the Pythagorean nano closure is a Kuratowski closure operator.  ... 
doi:10.35940/ijrte.e6477.018520 fatcat:37gv6up7prg2za5ckri7uqyafm
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