## The Kuratowski closure-complement theorem

Venue: | New Zealand Journal of Mathematics |

Citations: | 3 - 0 self |

### BibTeX

@ARTICLE{Gardner_thekuratowski,

author = {B. J. Gardner and M. Jackson},

title = {The Kuratowski closure-complement theorem},

journal = {New Zealand Journal of Mathematics},

year = {},

pages = {2008}

}

### OpenURL

### Abstract

The Kuratowski Closure-Complement Theorem 1.1. [29] If (X,T) is a topo-logical space and A ⊆ X then at most 14 sets can be obtained from A by taking closures and complements. Furthermore there is a space in which this bound is attained.

### Citations

754 | An Introduction to Modal Logic - Hughes, Cresswell - 1968 |

167 |
The theory of Representations of Boolean algebras
- Stone
(Show Context)
Citation Context ... (X, T) has fewer identifications than that of (Y, U). �sTHE KURATOWSKI CLOSURE-COMPLEMENT THEOREM 15 In the coming subsections we will also find it useful to recall some basic facts of Stone duality =-=[45]-=-. We direct the reader to Halmos [20] for an easy introduction and proofs of the statements below. (Note that our notation is different to that in [20].) For a Boolean algebra B, let B ∗ denote the se... |

121 |
The algebra of topology
- McKinsey, Tarski
- 1944
(Show Context)
Citation Context ...= [0, 1/3] ∩ C. The example of a 14-set in the Cantor space gives us the following result. Proposition 2.11. The Cantor space is a full Kuratowski space. This result also follows from Theorem 5.10 in =-=[32]-=-. Aull comments that the rationals given the usual subspace topology from the reals is a full Kuratowski space. 3. Stability under topological constructions We now examine what kind of properties of t... |

75 |
Lectures on Boolean Algebras
- Halmos
- 1963
(Show Context)
Citation Context ...n that of (Y, U). �sTHE KURATOWSKI CLOSURE-COMPLEMENT THEOREM 15 In the coming subsections we will also find it useful to recall some basic facts of Stone duality [45]. We direct the reader to Halmos =-=[20]-=- for an easy introduction and proofs of the statements below. (Note that our notation is different to that in [20].) For a Boolean algebra B, let B ∗ denote the set of all ultrafilters of B given a to... |

26 |
Elements of Mathematics. General Topology. Part 1
- Bourbaki
- 1966
(Show Context)
Citation Context ...g rise to any of the equivalence patterns are given in Table 2.1. The final column gives an example with the corresponding pattern. Here we let A = (0, 1), B = (1, 2), C = {3}, D = [4, 5] and E = Q ∩ =-=[6, 7]-=-, taken as subsets of the reals given the usual topology. For those equivalences not realised by any subset of a connected space, we use subsets of the Cantor space C. Let F = [0, 1/4) ∩ C, G = { n 3m... |

26 |
A problem of set theoretic topology
- Hewitt
- 1943
(Show Context)
Citation Context ... recall some definitions. Definition 2.2. (i) A space is extremally disconnected if the closure of any open set is open [46]. (ii) A space is resolvable if it contains a dense set with empty interior =-=[23]-=-. (iii) A space is an open unresolvable space if no open subspace is resolvable [2]. (iv) A space is a partition space if its open sets form a Boolean algebra [44, Example 5]. We note that many texts ... |

19 |
The lattice of modal logics: an algebraic investigation
- Blok
- 1980
(Show Context)
Citation Context ... Finding a subset of the reals with this property is an exercise is some textbooks [27, p. 57]. As shown in [44, Example 32, part 9], one example is the set {1/n | n ∈ N} ∪ (2, 3) ∪ (3, 4) ∪ {4 1 } ∪ =-=[5, 6]-=- ∪ {x | x is rational and 7 ≤ x < 8} 2 (but in fact only the 2 nd , 3 rd , 4 th and 6 th components are necessary). This completes the proof. ✷ An equivalent formulation of the Kuratowski closure-comp... |

12 | Topology via Logic. Cambridge Tracts - Vickers - 1989 |

3 |
Classification of topological spaces
- Aull
- 1967
(Show Context)
Citation Context ...the closure of any open set is open [46]. (ii) A space is resolvable if it contains a dense set with empty interior [23]. (iii) A space is an open unresolvable space if no open subspace is resolvable =-=[2]-=-. (iv) A space is a partition space if its open sets form a Boolean algebra [44, Example 5]. We note that many texts require that an extremally disconnected space satisfy the additional condition of b... |

3 |
Kuratowski numbers, in Application of functional analysis
- Chagrov
- 1982
(Show Context)
Citation Context ...s by their Kuratowski monoid appears to be the most natural of the possible ways of making use of the Kuratowski theorem. Despite this, the approach appears to have only been considered by Chagrov in =-=[10]-=-, a paper that is not readily available, and not available at all in English. Furthermore, the proof that the various Kuratowski monoids are the only possible Kuratowski monoids, which is not quite tr... |

3 |
A generalized closure and complement phenomenon
- Peleg
- 1984
(Show Context)
Citation Context .... See the discussion of Kuratowski boundary problems in Subsection 4.1 for some of the results of [43]. The Kuratowski-type behaviour of generalised notions of closure are rediscovered again in Peleg =-=[36]-=-. Peleg also considers several interesting concrete examples such as the complement operator and Kleene closure on formal languages: the maximum number of languages obtainable is 14 and it is commente... |

3 |
Quelques notions fondamentales de l’Analysis Situs au point de vue de l’Algèbra de la Logique, Fund
- Zarycki
- 1927
(Show Context)
Citation Context ...aviour of f in conjunction with subsets of {a, b, i} are neither well known nor easily accessible. There appear to be three places where the problem is studied in some detail. The first is in Zarycki =-=[50]-=-, where the Kuratowski {a, f}-problem is shown to have a solution of 6. This is extended substantially by the work of Soltan [43]. In fact a slightly more general notion than topological closure is co... |

2 |
Remarks on finite topological spaces
- Anusiak, Shum
- 1971
(Show Context)
Citation Context ...full Kuratowski space on seven points is also given in both papers and in [22] it is shown that there are exactly 5 non-homeomorphic spaces on 7 points with k number 14. These results are extended in =-=[1]-=- where it shown that k((X, T)) ≤ 2|X| for all |X| ≤ 7 (and that this bound can be achieved in every case). Furthermore, for all n ≤ 7, the number of non-homeomorphic n-point spaces with k-number 2n is... |

2 |
The free closure algebra on finitely many generators
- Blok
- 1977
(Show Context)
Citation Context ... Finding a subset of the reals with this property is an exercise is some textbooks [27, p. 57]. As shown in [44, Example 32, part 9], one example is the set {1/n | n ∈ N} ∪ (2, 3) ∪ (3, 4) ∪ {4 1 } ∪ =-=[5, 6]-=- ∪ {x | x is rational and 7 ≤ x < 8} 2 (but in fact only the 2 nd , 3 rd , 4 th and 6 th components are necessary). This completes the proof. ✷ An equivalent formulation of the Kuratowski closure-comp... |

2 |
A further note on closure and interior operators
- Chapman
- 1962
(Show Context)
Citation Context ...xamined. These functions, called line functions are then abstracted and it is shown that an arbitrary set X and with a pair of line functions acting on the subsets of X determines a topology on X. In =-=[12]-=- Chapman presents a deeper investigation into which Kuratowski operators can act differently on subsets of topological spaces. For every pair (o1, o2) of the 7 operators in Figure 1.1, a description i... |

2 |
The Kuratowski closure-complement problem
- Fife
- 1991
(Show Context)
Citation Context ...In general we say that an operator o is isotonic if X ⊆ Y ⇒ o(X) ⊆ o(Y ) and is idempotent if o2 = o. With these ideas in hand it is now quite easy to give a proof of the Kuratowski theorem (see also =-=[29, 21, 30, 15, 8]-=-). 2000 Mathematics Subject Classification. 54A05, 54B99, 54G99, 06F99. 1s2 B. J. GARDNER AND MARCEL JACKSON We present this proof for completeness. Proof of the Kuratowski Closure-Complement Theorem.... |

2 |
Kuratowski’s Closure theorem, Nieuw
- Hammer
- 1960
(Show Context)
Citation Context ...In general we say that an operator o is isotonic if X ⊆ Y ⇒ o(X) ⊆ o(Y ) and is idempotent if o2 = o. With these ideas in hand it is now quite easy to give a proof of the Kuratowski theorem (see also =-=[29, 21, 30, 15, 8]-=-). 2000 Mathematics Subject Classification. 54A05, 54B99, 54G99, 06F99. 1s2 B. J. GARDNER AND MARCEL JACKSON We present this proof for completeness. Proof of the Kuratowski Closure-Complement Theorem.... |

2 |
Sur l’opération A de l’Analysis Situs, Fund
- Kuratowski
- 1922
(Show Context)
Citation Context ...In general we say that an operator o is isotonic if X ⊆ Y ⇒ o(X) ⊆ o(Y ) and is idempotent if o2 = o. With these ideas in hand it is now quite easy to give a proof of the Kuratowski theorem (see also =-=[29, 21, 30, 15, 8]-=-). 2000 Mathematics Subject Classification. 54A05, 54B99, 54G99, 06F99. 1s2 B. J. GARDNER AND MARCEL JACKSON We present this proof for completeness. Proof of the Kuratowski Closure-Complement Theorem.... |

2 |
Characterisation of Kuratowski 14-sets
- Langford
- 1971
(Show Context)
Citation Context |

2 |
On the boundary of Kuratowski 14-sets in connected spaces
- Shum
- 1984
(Show Context)
Citation Context ...is a particular case. A description of typical flavour is that A = bi(A) if and only if A is closed and is the union of an open set and a set with empty interior in the subspace on A. Papers [30] and =-=[39]-=- primarily concern the characterisation of 14-sets. The main theorem in [30] gives five independent conditions on a set A, which together are equivalent to A being a 14-set. This is then used to show ... |

2 |
An illumination problem
- Soltan
- 1994
(Show Context)
Citation Context ...satisfying f ≥ id. Some problems involving generalised closures that are posed in [21] are solved in [13] and [40]. Further investigations into generalised closures can be found in the work of Soltan =-=[42, 43]-=-. In [42], relationships between the number of sets obtainable using (generalised) closure and complement and the number obtainable using generalised closure and its corresponding interior are investi... |

1 |
Advanced Problem 5569
- Baron
- 1968
(Show Context)
Citation Context ...lent to A being a 14-set. This is then used to show that for every 14-set A ⊆ R with the usual topology there is an interval I in which both A and a(A) are dense (this solved a problem posed by Baron =-=[3]-=-). In [39] there is another characterisation of 14-sets under the additional assumption that the topological space be connected. Several variations of the notion of boundary are introduced and it is s... |

1 |
The Kuratowski closure-complement problem
- Berman, Jordan
- 1975
(Show Context)
Citation Context ...en b(A) is clopen. � It is also easy to see that partition space has k-number 6 if and only if it is neither discrete nor anti-discrete. A further result relating spaces to their k-number is given in =-=[4]-=- and [22] where it is shown that k((X, T)) = 14 implies that |X| ≥ 7; an example of a full Kuratowski space on seven points is also given in both papers and in [22] it is shown that there are exactly ... |

1 |
The fourteen subsets problem: interiors closures and complements
- Brandsma
- 2003
(Show Context)
Citation Context |

1 | Problem - Buchman - 1986 |

1 |
An extension of the Kuratowski closure and complementation problem
- Chapman
- 1962
(Show Context)
Citation Context ...n topological spaces. The earliest work (other than that of Kuratowski) concerning the possible behaviour of the topological operators on individual sets appears to be that of Levine [31] and Chapman =-=[11]-=-. Levine gives characterisations of subsets of spaces on which the interior and closure operators commute. In [11], two functions are considered which associate with each subset A of X, a positive int... |

1 |
On the composition semigroup generated by complement and an expansive function, Nieuw
- Coleman
- 1967
(Show Context)
Citation Context ..., a closure operator on the subsets of a set X is defined to be an isotonic, idempotent operator f satisfying f ≥ id. Some problems involving generalised closures that are posed in [21] are solved in =-=[13]-=- and [40]. Further investigations into generalised closures can be found in the work of Soltan [42, 43]. In [42], relationships between the number of sets obtainable using (generalised) closure and co... |

1 |
The standard variety of operators of a variety, Algebra Universalis
- Comer, Johnson
- 1972
(Show Context)
Citation Context ... combinations of these, the largest being the operator HSP. Moreover, the bound of 18 is achieved over families of groups. A survey of results concerning operators obtained from {H, S, P} is given in =-=[14]-=-. In [35] analogous questions are investigated for operator monoids generated by other sets of class operators—such as {H, S, P, PS} (where PS indicates subdirect product). The particular case of {H, ... |

1 |
Operations on binary relations
- Fishburn
- 1978
(Show Context)
Citation Context ...ransitive closures and reflexive closures is 42. The transitive closure and complement result is further developed in [17]. A larger class still of operators on relations is considered by Fishburn in =-=[16]-=-. Here it is shown that the maximum number of relations obtainable from a single binary relation by applying the operators of asymmetrisation, complementation, dualisation, symmetrisation, and transit... |

1 |
On closures verifying that the interior of a closed element is closed
- Garel, Olivier
- 1995
(Show Context)
Citation Context ...ssible number of relations obtained from a given relation by taking complements, transitive closures and reflexive closures is 42. The transitive closure and complement result is further developed in =-=[17]-=-. A larger class still of operators on relations is considered by Fishburn in [16]. Here it is shown that the maximum number of relations obtainable from a single binary relation by applying the opera... |

1 |
Closure and interior in finite topological spaces
- Herda, Metzler
- 1966
(Show Context)
Citation Context ...is clopen. � It is also easy to see that partition space has k-number 6 if and only if it is neither discrete nor anti-discrete. A further result relating spaces to their k-number is given in [4] and =-=[22]-=- where it is shown that k((X, T)) = 14 implies that |X| ≥ 7; an example of a full Kuratowski space on seven points is also given in both papers and in [22] it is shown that there are exactly 5 non-hom... |

1 |
Closure semilattices, Algebra Universalis
- Jackson
(Show Context)
Citation Context ...maller interior. Hence a minimum of 14 points is required. � The Kuratowski {b, ∧} problem. Clearly if we start from a single set, only 2 sets can be obtained by applying closure and intersection. In =-=[25]-=- however, a topological space is constructed with two subsets A and B, for which infinitely many subsets can be obtained by taking closures and intersections of the sets A and B; furthermore, the poss... |

1 |
Semilattice pseudocomplemented semigroups
- Jackson, Stokes
(Show Context)
Citation Context ...ological space: it assigns to each subset A of a topological space (X, T), the largest member B of T for which A ∩ B = ∅. A system of algebraic structures related to this operator has been studied in =-=[26]-=- and there it is shown that the number of sets obtainable from a single subset of a topological space using ia and ∧ is equal to 10. The standard example of a Kuratowski 14-set given in [44, Example 3... |

1 |
The Kuratowski closure problem in the topology of convexity
- Koenen
- 1966
(Show Context)
Citation Context ...number of languages obtainable is 14 and it is commented that this bound is achieved by the language {a, aab, bbb}. A natural generalised closure operator called the convex hull is examined by Koenen =-=[28]-=-. This operator associates with each subset S of R n the intersection of all convex sets containing S. It is shown that at most 10 sets can be produced from a subset of R n by taking convex hulls and ... |

1 |
On the commutativity of the closure and interior operators in topological spaces
- Levine
- 1961
(Show Context)
Citation Context ...numbers of sets in topological spaces. The earliest work (other than that of Kuratowski) concerning the possible behaviour of the topological operators on individual sets appears to be that of Levine =-=[31]-=- and Chapman [11]. Levine gives characterisations of subsets of spaces on which the interior and closure operators commute. In [11], two functions are considered which associate with each subset A of ... |

1 |
Closure, interior and union in finite topological spaces
- Moser
- 1977
(Show Context)
Citation Context ...ve used larger points to indicate the even Kuratowski operators). For example, the left diagram shows that b(bi ∧ ib) = bi, because bi is the smallest closed operator above bi ∧ ib in the diagram. In =-=[33]-=- it is shown that the smallest space containing a subset on which all 13 operators in Figure 4.1 (a) act differently has 9 points. Analogously to [1], the maximum possible number of distinct operators... |

1 |
A generalization of the Kuratowski closure-complement problem
- Moslehian, Tavallaii
- 1995
(Show Context)
Citation Context ... in the table can be achieved (recall that the closure in [43] is not necessarily topological). The third place where boundaries are used in conjunction with closures, interiors and complements is in =-=[34]-=-, where the first 5 values of the above table are again obtained. For each of these cases, a description is given of what properties a subset of a space must have in order to achieve this bound (exten... |

1 |
Finiteness of semigroups of operators in universal algebras
- Nelson
- 1967
(Show Context)
Citation Context ...ions of these, the largest being the operator HSP. Moreover, the bound of 18 is achieved over families of groups. A survey of results concerning operators obtained from {H, S, P} is given in [14]. In =-=[35]-=- analogous questions are investigated for operator monoids generated by other sets of class operators—such as {H, S, P, PS} (where PS indicates subdirect product). The particular case of {H, S, P, PS}... |

1 |
On some operations on classes of algebras, Algebra Universalis 2
- Pigozzi
- 1972
(Show Context)
Citation Context ...lgebras (that no more than 14 sets can be obtained in this way follows from the proof of the Kuratowski theorem). As a final example of interest (also of algebraic nature) we note a result of Pigozzi =-=[37]-=-. Here the standard class operators of universal algebra are considered; that is, closure operators on classes of algebras of a particular type. There are three operators: S, which corresponds to taki... |

1 |
Variations on Kuratowski’s 14-set theorem, online manuscript
- Sherman
- 2004
(Show Context)
Citation Context ...to partition spaces (see [24, Chapter 3]). After this article was initially prepared for publication, the authors became aware of an article by David Sherman posted on the online archive Math Archive =-=[38]-=- where similar results to some of the ones in this subsection are surveyed. The Kuratowski {a, b, ∨}- and {a, b, ∧}-problem. This problem is solved by Kuratowski in his original paper (see page 197 of... |

1 |
Closure functions on the set of positive integers
- Shum
- 1996
(Show Context)
Citation Context ...re operator on the subsets of a set X is defined to be an isotonic, idempotent operator f satisfying f ≥ id. Some problems involving generalised closures that are posed in [21] are solved in [13] and =-=[40]-=-. Further investigations into generalised closures can be found in the work of Soltan [42, 43]. In [42], relationships between the number of sets obtainable using (generalised) closure and complement ... |

1 |
Advanced Problem 5996
- Smith
- 1974
(Show Context)
Citation Context ...rators t0, t1, . . . produce distinct sets when applied to the subset {1, 3, 5, . . . } in B. The Kuratowski {b, i, ∨} problem. This problem is the topic of American Mathematical Monthly Problem 5996 =-=[41]-=- (a solution is given in [49]): at most 13 sets can be obtained from using closure, intersection and union from a subset of a topological space. (This result is also rediscovered in [38].) The Kuratow... |

1 |
Problems of Kuratowski type
- Soltan
- 1982
(Show Context)
Citation Context ...s where the problem is studied in some detail. The first is in Zarycki [50], where the Kuratowski {a, f}-problem is shown to have a solution of 6. This is extended substantially by the work of Soltan =-=[43]-=-. In fact a slightly more general notion than topological closure is considered in [43] (see Subsection 4.2 for other variants of this type), but with only minor modifications one obtains the followin... |

1 |
Algebraic characterisations of special Boolean rings, Fund
- Stone
- 1937
(Show Context)
Citation Context ...n Kuratowski operators. We now give slightly more structural descriptions. First recall some definitions. Definition 2.2. (i) A space is extremally disconnected if the closure of any open set is open =-=[46]-=-. (ii) A space is resolvable if it contains a dense set with empty interior [23]. (iii) A space is an open unresolvable space if no open subspace is resolvable [2]. (iv) A space is a partition space i... |

1 | On the partially ordered monoid generated by the operators - Tasić |

1 |
Solution to Problem 5996
- Yu
- 1978
(Show Context)
Citation Context ...distinct sets when applied to the subset {1, 3, 5, . . . } in B. The Kuratowski {b, i, ∨} problem. This problem is the topic of American Mathematical Monthly Problem 5996 [41] (a solution is given in =-=[49]-=-): at most 13 sets can be obtained from using closure, intersection and union from a subset of a topological space. (This result is also rediscovered in [38].) The Kuratowski {b, i, ∧} problem follows... |

1 |
Kuratowski’s Closure theorem
- Hammer
- 1960
(Show Context)
Citation Context ...o. With these ideas in hand it is now quite easy to give a proof of the Kuratowski theorem (see also 2000 Mathematics Subject Classification 54A05, 54B99, 54G99, 06F99. 10 B.J. GARDNER AND M. JACKSON =-=[29, 21, 30, 15, 8]-=-). We present this proof for completeness. Proof of the Kuratowski Closure-Complement Theorem. First note that the operator a satisfies a2 = id (where id is the identity operator), while b is idempote... |

1 |
On some operations on classes of algebras
- Pigozzi
- 1972
(Show Context)
Citation Context ...lgebras (that no more than 14 sets can be obtained in this way follows from the proof of the Kuratowski theorem). As a final example of interest (also of algebraic nature) we note a result of Pigozzi =-=[37]-=-. Here the standard class operators of universal algebra are considered; that is, closure operators on classes of algebras of a particular type. There are three operators: S, which corresponds to taki... |

1 | Variations on Kuratowski’s 14-set theorem
- Sherman
(Show Context)
Citation Context ...perators will be called the Kuratowski O-problem. An operator o derived by composing members of some set O of operators will be called an O-operator. The reader is also directed to the recent article =-=[38]-=- by David Sherman, which has some overlap with this section and provides an excellent introduction to the Kuratowski Closure-Complement Theorem. 4.1. Topological variants. In this section we will prim... |

1 | On the partially ordered monoid generated by the operators H,S, P, Ps on classes of algebras - Tasić |