FUMIE NAKAOKA AND NOBUYUKI ODA
Received 20 December 2005; Revised 28 May 2006; Accepted 6 August 2006
Some properties of minimal closed sets and maximal closed sets are obtained, which are dual concepts of maximal open sets and minimal open sets, respectively. Common properties of minimal closed sets and minimal open sets are clarified; similarly, common properties of maximal closed sets and maximal open sets are obtained. Moreover, inter-relations of these four concepts are studied.
Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
1. Introduction
Some properties of minimal open sets and maximal open sets are studied in [1,2]. In this paper, we define dual concepts of them, namely, maximal closed set and minimal closed set. These four types of subsets appear in finite spaces, for example. More generally, min-imal open sets and maxmin-imal closed sets appear in locally finite spaces such as the digital line. Minimal closed sets and maximal open sets appear in cofinite topology, for example. We have to study these four concepts to understand the relations among their properties. Some of the results in [1,2] can be dualized using standard techniques of general topol-ogy. But we have to study the “dual results” carefully to understand the duality which we propose in this paper and in [2]. For example, considering the interrelation of these four concepts, we see that some results in [1,2] can be generalized further.
In [1] we called a nonempty open setUofXa minimal open set if any open set which is contained inU is∅orU. But to consider duality, we have to consider only “proper nonempty open setU of X,” as in the following definitions: a proper nonempty open subsetUofXis said to be a minimal open set if any open set which is contained inU is ∅orU. A proper nonempty open subsetU ofXis said to be a maximal open set if any open set which containsUisXorU. In this paper, we will use the following definitions. A proper nonempty closed subsetFofXis said to be a minimal closed set if any closed set which is contained inFis∅orF. A proper nonempty closed subsetFofXis said to be a maximal closed set if any closed set which containsFisXorF.
Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 18647, Pages1–8
LetFbe a subset of a topological spaceX. Then the following duality principle holds: (1)Fis a minimal closed set if and only ifX−Fis a maximal open set;
(2)Fis a maximal closed set if and only ifX−Fis a minimal open set.
In Sections2and3, we will show some results on minimal closed sets and maximal closed sets. InSection 4, we will obtain some further results on minimal open sets and maximal open sets.
The symbolΛ\Γmeans difference of index sets, namely,Λ\Γ=Λ−Γ, and the cardi-nality of a setΛis denoted by|Λ|in the following arguments. A subsetMof a topological spaceXis called a pre-open set ifM⊂Int(Cl(M)) and a subsetMis called a pre-closed set ifX−Mis a pre-open set.
2. Minimal closed sets
In this section, we prove some results on minimal closed sets.
The following result shows that a fundamental result of [1, Lemma 2.2] also holds for closed sets; it is the dual result of [2, Lemma 2.2].
Lemma 2.1. (1) LetFbe a minimal closed set andNa closed set. ThenF∩N= ∅orF⊂N. (2) LetFandSbe minimal closed sets. ThenF∩S= ∅orF=S.
The proof ofLemma 2.1is omitted, since it is obtained by an argument similar to the proof of [1, Lemma 2.2]. Now, we generalize the dual result of [2, Theorem 2.4].
Theorem 2.2. LetFandFλbe minimal closed sets for any elementλofΛ. (1) IfF⊂λ∈ΛFλ, then there exists an elementλofΛsuch thatF=Fλ. (2) IfF=Fλfor any elementλofΛ, then (λ∈ΛFλ)∩F= ∅.
Proof. (1) SinceF⊂λ∈ΛFλ, we getF=F∩(λ∈ΛFλ)=λ∈Λ(F∩Fλ). IfF=Fλfor any elementλofΛ, thenF∩Fλ= ∅for any elementλofΛbyLemma 2.1(2), hence we have ∅ =λ∈Λ(F∩Fλ)=F. This contradicts our assumption thatFis a minimal closed set. Thus there exists an elementλofΛsuch thatF=Fλ.
(2) If (λ∈ΛFλ)∩F= ∅, then there exists an elementλofΛsuch thatFλ∩F= ∅. By
Lemma 2.1(2), we haveFλ=F, which is a contradiction.
Corollary 2.3. LetFλbe a minimal closed set for any elementλofΛandFλ=Fμfor any elementsλandμofΛwithλ=μ. IfΓis a proper nonempty subset ofΛ, then (λ∈Λ\ΓFλ)∩ (γ∈ΓFγ)= ∅.
The following result is a generalization of the dual result of [2, Theorem 2.5].
Theorem 2.4. LetFλandFγbe minimal closed sets for any elementλofΛandγof Γ. If there exists an elementγofΓsuch thatFλ=Fγfor any elementλofΛ, thenγ∈ΓFγ⊂λ∈ΛFλ. Proof. Suppose that an elementγofΓsatisfiesFλ=Fγfor any elementλofΛ. Ifγ∈ΓFγ⊂
λ∈ΛFλ, then we getFγ⊂λ∈ΛFλ. ByTheorem 2.2(1), there exists an elementλof Λ
such thatFγ=Fλ, which is a contradiction.
Theorem 2.5. LetFλbe a minimal closed set for any elementλofΛandFλ=Fμ for any elementsλandμofΛwithλ=μ. IfΓis a proper nonempty subset ofΛ, thenγ∈ΓFγ
λ∈ΛFλ.
Proof. Let κ be any element of Λ\Γ. Then Fκ∩(γ∈ΓFγ)=γ∈Γ(Fκ∩Fγ)= ∅ and Fκ∩(λ∈ΛFλ)=λ∈Λ(Fκ∩Fλ)=Fκ. Ifγ∈ΓFγ=λ∈ΛFλ, then we have∅ =Fκ. This contradicts our assumption thatFκis a minimal closed set. The following theorem is the dual result of [2, Theorem 4.2] and is the key to the proof ofTheorem 2.7, which is closely connected with [2, Theorem 4.7].
Theorem 2.6. Assume that|Λ| ≥2 and letFλbe a minimal closed set for any elementλof ΛandFλ=Fμ for any elementsλandμofΛwithλ=μ. ThenFμ⊂X−λ∈Λ\{μ}Fλand henceλ∈Λ\{μ}Fλ=Xfor any elementμofΛ.
Proof. ByCorollary 2.3, we have the result.
Theorem 2.7 (recognition principle for minimal closed sets). Assume that|Λ| ≥2 and letFλbe a minimal closed set for any elementλofΛandFλ=Fμfor any elementsλandμof Λwithλ=μ. Then for any elementμofΛ,
Fμ=
λ∈Λ Fλ
∩
X−
λ∈Λ\{μ}
Fλ
. (2.1)
Proof. Letμbe an element ofΛ. ByTheorem 2.6, we have
λ∈ΛFλ
∩
X−
λ∈Λ\{μ}
Fλ
=
λ∈Λ\{μ}
Fλ
∪Fμ
∩
X−
λ∈Λ\{μ}
Fλ
=
λ∈Λ\{μ}
Fλ
∩
X−
λ∈Λ\{μ}
Fλ
∪
Fμ∩
X−
λ∈Λ\{μ}
Fλ
=Fμ∩
X−
λ∈Λ\{μ}
Fλ
=Fμ.
(2.2)
As an application ofTheorem 2.7, we prove the following result which is the dual result of [2, Theorem 4.8].
Theorem 2.8. LetFλbe a minimal closed set for any elementλof a finite setΛandFλ=Fμ for any elementsλandμofΛwithλ=μ. Ifλ∈ΛFλis an open set, thenFλis an open set for any elementλofΛ.
Proof. Let μ be an element of Λ. By Theorem 2.7, we have Fμ=(λ∈ΛFλ)∩(X−
λ∈Λ\{μ}Fλ)=(λ∈ΛFλ)∩(λ∈Λ\{μ}(X−Fλ)). By our assumption, it is seen that
As an immediate consequence ofTheorem 2.6, we have the following result which is the dual of [2, Theorem 4.9].
Theorem 2.9. Assume that|Λ| ≥2 and letFλbe a minimal closed set for any elementλof ΛandFλ=Fμfor any elementsλandμofΛwithλ=μ. Ifλ∈ΛFλ=X, then{Fλ|λ∈Λ} is the set of all minimal closed sets ofX.
LetᏲ= {Fλ|λ∈Λ}be a set of minimal closed sets. We call∪Ᏺ=λ∈ΛFλthe corad-ical ofᏲ. The following result about coradical is obtained by [2, Theorem 4.13].
Theorem 2.10. LetFλbe a minimal closed set for any elementλofΛandFλ=Fμfor any elementsλandμofΛwithλ=μ. Ifλ∈ΛFλis an open set, thenλ∈Λ\{μ}Fλis an open set for any elementμofΛ.
Theorem 2.11. LetFbe a minimal closed set. Then, Int(F)=For Int(F)= ∅.
Proof. If we putU=X−Fin [2, Theorem 3.5], then we have the result. IfSis a proper subset of a minimal closed set, then Int(S)= ∅and henceSis a pre-closed set (cf. [2, Corollary 3.7]).
Theorem 2.12. Let Fλ be a minimal closed set for any element λ of a finite set Λ. If Int(λ∈ΛFλ)= ∅, then there exists an elementλofΛsuch that Int(Fλ)=Fλ.
Proof. Since Int(λ∈ΛFλ)= ∅, there exists an elementxof Int(λ∈ΛFλ). Then there ex-ists an open set U such that x∈U⊂λ∈ΛFλ and hence there exists an elementμ of Λsuch thatx∈Fμ. By Theorems2.6and2.7,x∈U∩(X−λ∈Λ\{μ}Fλ)⊂(λ∈ΛFλ)∩ (X−λ∈Λ\{μ}Fλ)=Fμ. SinceU∩(X−λ∈Λ\{μ}Fλ) is an open set, we see thatxis an element of Int(Fμ) and hence Int(Fμ)= ∅. Therefore there exists an elementμofΛsuch
that Int(Fμ)=FμbyTheorem 2.11.
Remark 2.13. IfΛis an infinite set, thenTheorem 2.12does not hold. For example, see [2, Example 5.3].
Theorem 2.14 (the law of interior of coradical). LetFλ be a minimal closed set for any elementλof a finite setΛandFλ=Fμfor any elementsλandμofΛwithλ=μ. LetΓbe a subset ofΛsuch that
IntFλ=Fλ for anyλ∈Γ,
IntFλ= ∅ for anyλ∈Λ\Γ. (2.3) Then Int(λ∈ΛFλ)=λ∈ΓFλ(= ∅ifΓ= ∅).
Proof. By [2, Theorem 5.4], we have the result.
3. Maximal closed sets
The following lemma is the dual ofLemma 2.1.
LetFλbe a maximal closed set for any elementλofΛ. LetᏲ= {Fλ|λ∈Λ}. We call
Ᏺ=λ∈ΛFλthe radical ofᏲ.
Corollary 3.2. LetFandFλbe maximal closed sets for any elementλofΛ. IfF=Fλfor any elementλofΛ, then (λ∈ΛFλ)∪F=X.
Proof. ByLemma 3.1(2), we have the result.
Theorem 3.3. LetFandFλbe maximal closed sets for any elementλofΛ. Ifλ∈ΛFλ⊂F, then there exists an elementλofΛsuch thatF=Fλ.
Proof. Sinceλ∈ΛFλ⊂F, we getF=F∪(λ∈ΛFλ)=λ∈Λ(F∪Fλ). IfF∪Fλ=X for any elementλofΛ, then we haveX=λ∈Λ(F∪Fλ)=F. This contradicts our assumption thatFis a maximal closed set. Then there exists an elementλofΛsuch thatF∪Fλ=X.
ByLemma 3.1(2), we have the result.
Corollary 3.4. LetFλandFγ be maximal closed sets for any elementsλ∈Λandγ∈Γ. Ifγ∈ΓFγ⊂λ∈ΛFλ, then for any elementλofΛ, there exists an elementγ∈Γsuch that Fλ=Fγ.
Corollary 3.5. LetFα,Fβ, andFγbe maximal closed sets which are different from each other. Then
Fα∩Fβ⊂Fα∩Fγ. (3.1)
Theorem 3.6. LetFλ be a maximal closed set for any elementλofΛandFλ=Fμ for any elementsλandμofΛwithλ=μ. IfΓis a proper nonempty subset ofΛ, thenλ∈Λ\ΓFλ⊂
γ∈ΓFγ⊂λ∈Λ\ΓFλ.
Proof. Letγ be any element ofΓ. Ifλ∈Λ\ΓFλ⊂Fγ, then we getFλ=Fγ for someλ∈ Λ\ΓbyTheorem 3.3. This contradicts our assumption. Therefore we haveλ∈Λ\ΓFλ⊂
γ∈ΓFγ. On the other hand, sinceγ∈ΓFγ=γ∈Λ\(Λ\Γ)Fγ⊂
λ∈Λ\ΓFλ, we haveγ∈ΓFγ⊂
λ∈Λ\ΓFλ.
Theorem 3.7. LetFλ be a maximal closed set for any elementλofΛandFλ=Fμ for any elementsλandμofΛwithλ=μ. IfΓis a proper nonempty subset ofΛ, thenλ∈ΛFλ
γ∈ΓFγ.
Proof. Letκbe any element ofΛ\Γ. ThenFκ∪(γ∈ΓFγ)=γ∈Γ(Fκ∪Fγ)=XandFκ∪ (λ∈ΛFλ)=λ∈Λ(Fκ∪Fλ)=Fκ. Ifγ∈ΓFγ=λ∈ΛFλ, then we haveX=Fκ. This
contra-dicts our assumption thatFκis a maximal closed set.
Theorem 3.8. Assume that|Λ| ≥2 and letFλbe a maximal closed set for any elementλof ΛandFλ=Fμ for any elementsλandμofΛwithλ=μ. ThenX−λ∈Λ\{μ}Fλ⊂Fμand henceλ∈Λ\{μ}Fλ= ∅for any elementμofΛ.
Proof. Letμbe any element ofΛ. ByLemma 3.1(2), we haveX−Fμ⊂Fλfor any element λofΛwithλ=μ. ThenX−Fμ⊂λ∈Λ\{μ}Fλ. Therefore we haveX−λ∈Λ\{μ}Fλ⊂Fμ.
Theorem 3.9 (decomposition theorem for maximal closed sets). Assume that |Λ| ≥2 and letFλbe a maximal closed set for any elementλofΛandFλ=Fμfor any elementsλand μofΛwithλ=μ. Then for any elementμofΛ,
Fμ=
λ∈ΛFλ
∪
X− λ∈Λ\{μ}
Fλ
. (3.2)
Proof. Letμbe an element ofΛ. ByTheorem 3.8, we have
λ∈ΛFλ
∪
X− λ∈Λ\{μ}
Fλ
=
λ∈Λ\{μ}
Fλ
∩Fμ
∪
X− λ∈Λ\{μ}
Fλ
=
λ∈Λ\{μ}
Fλ
∪
X− λ∈Λ\{μ}
Fλ
∩
Fμ∪
X− λ∈Λ\{μ}
Fλ
=Fμ∪
X− λ∈Λ\{μ}
Fλ
=Fμ.
(3.3)
Theorem 3.10. LetFλbe a maximal closed set for any elementλofΛandFλ=Fμfor any elementsλandμofΛwithλ=μ. Ifλ∈ΛFλ is an open set, thenFλis an open set for any elementλofΛ.
Proof. By Theorem 3.9, we have Fμ =(λ∈ΛFλ)∪(X −λ∈Λ\{μ}Fλ)=(λ∈ΛFλ)∪ (λ∈Λ\{μ}(X−Fλ)). Then λ∈Λ\{μ}(X−Fλ) is an open set. HenceFμ is an open set by
our assumption.
Theorem 3.11. Assume that|Λ| ≥2 and letFλbe a maximal closed set for any elementλof ΛandFλ=Fμfor any elementsλandμofΛwithλ=μ. Ifλ∈ΛFλ= ∅, then{Fλ|λ∈Λ} is the set of all maximal closed sets ofX.
Proof. If there exists another maximal closed set Fν of X which is not equal to Fλ for any elementλof Λ, then∅ =λ∈ΛFλ=λ∈(Λ∪{ν})\{ν}Fλ. ByTheorem 3.8, we get
λ∈(Λ∪{ν})\{ν}Fλ= ∅. This contradicts our assumption.
Theorem 3.12. Assume that|Λ| ≥2 and letFλ be a maximal closed set for any element λofΛandFλ=Fμ for any elementsλ andμofΛwithλ=μ. If Int(λ∈ΛFλ)= ∅, then {Fλ|λ∈Λ}is the set of all maximal closed sets ofX.
Proof. If there exists another maximal closed setFofXwhich is not equal toFλfor any elementλ of Λ, thenX−F⊂λ∈ΛFλ byTheorem 3.8. It follows that Int(λ∈ΛFλ)⊃ Int(X−F)=X−F= ∅. This contradicts our assumption.
The proof of the following lemma is immediate and is omitted.
Proposition 3.14. LetFλbe a closed set for any elementλofΛandFλ∪Fμ=X for any elementsλandμofΛwithλ=μ. Ifλ∈ΛFλis an open set, thenλ∈Λ\{μ}Fλis an open set for any elementμofΛ.
Proof. Letμbe any element ofΛ. SinceFλ∪Fμ=Xfor any elementλofΛwithλ=μ, we haveFμ∪(λ∈Λ\{μ}Fλ)=λ∈Λ\{μ}(Fμ∪Fλ)=X. SinceFμ∩(λ∈Λ\{μ}Fλ)=λ∈ΛFλ is an open set by our assumption,λ∈Λ\{μ}Fλis an open set byLemma 3.13.
Theorem 3.15. LetFλbe a maximal closed set for any elementλofΛandFλ=Fμfor any elementsλandμofΛwithλ=μ. Ifλ∈ΛFλis an open set, thenλ∈Λ\{μ}Fλis an open set for any elementμofΛ.
Proof. ByLemma 3.1(2), we haveFλ∪Fμ=Xfor any elementsλandμofΛwithλ=μ.
ByProposition 3.14, we haveλ∈Λ\{μ}Fλis an open set.
4. Minimal open sets and maximal open sets
In this section, we record some results on minimal open sets and maximal open sets, which are not proved in [1,2]. The proofs are omitted since they are obtained by dual arguments.
Letᐁ= {Uλ|λ∈Λ}be a set of minimal open sets. We callᐁ=λ∈ΛUλthe corad-ical ofᐁ. By an argument similar toTheorem 2.2, we have the following result.
Theorem 4.1. LetUandUλbe minimal open sets for any elementλofΛ. (1) IfU⊂λ∈ΛUλ, then there exists an elementλofΛsuch thatU=Uλ. (2) IfU=Uλfor any elementλofΛ, then (λ∈ΛUλ)∩U= ∅.
Corollary 4.2. LetUλbe a minimal open set for any elementλofΛandUλ=Uμfor any elementsλandμofΛwithλ=μ. IfΓis a proper nonempty subset ofΛ, then (λ∈Λ\ΓUλ)∩ (γ∈ΓUγ)= ∅.
The following results are shown by the arguments similar to the proofs of Theorems
2.4,2.5, and2.6, respectively.
Theorem 4.3. LetUλandUγbe minimal open sets for any elementsλ∈Λandγ∈Γ. If there exists an elementγofΓsuch thatUλ=Uγ for any elementλofΛ, thenγ∈ΓUγ⊂
λ∈ΛUλ.
Theorem 4.4. LetUλ be a minimal open set for any elementλofΛandUλ=Uμ for any elementsλandμofΛwithλ=μ. IfΓis a proper nonempty subset ofΛ, thenγ∈ΓUγ
λ∈ΛUλ.
Theorem 4.5. Assume that|Λ| ≥2 and letUλbe a minimal open set for any elementλof ΛandUλ=Uμfor any elementsλandμofΛwithλ=μ. ThenUμ⊂X−λ∈Λ\{μ}Uλand henceλ∈Λ\{μ}Uλ=Xfor any elementμofΛ.
We obtain the following result for minimal open sets (cf. Theorems2.7and3.9).
withλ=μ. Then for any elementμofΛ,
Uμ=
λ∈Λ Uλ
∩
X−
λ∈Λ\{μ}
Uλ
. (4.1)
Theorem 4.7. LetUλbe a minimal open set for any elementλof any setΛandUλ=Uμfor any elementsλandμofΛwithλ=μ. Ifλ∈ΛUλis a closed set, thenUλis a closed set for any elementλofΛ(cf. Theorems2.8and3.10).
Theorem 4.8. Assume that|Λ| ≥2 and letUλbe a minimal open set for any elementλofΛ andUλ=Uμfor any elementsλandμofΛwithλ=μ. Ifλ∈ΛUλ=X, then{Uλ|λ∈Λ} is the set of all minimal open sets ofX(cf. Theorems2.9and3.11).
Theorem 4.9. Assume that|Λ| ≥2 and letUλ be a minimal open set for any elementλ ofΛandUλ=Uμ for any elementsλ andμof Λwith λ=μ. If Cl(λ∈ΛUλ)=X, then {Uλ|λ∈Λ}is the set of all minimal open sets ofX(cf. Theorem3.12).
Example 4.10 (the digital line). The digital line is the setZof the intergers equipped with the topologyτhaving a family of subsetsS= {{2k−1, 2k, 2k+ 1} |k∈Z}as a subbase. We consider a set of minimal open sets{Uk= {2k+ 1} |k∈Z}. Then Cl(k∈ZUk)= Cl({2k+ 1|k∈Z})=Zand hence{Uk= {2k+ 1} |k∈Z}is the set of all minimal open sets in (Z,τ).
Finally we consider maximal open sets. The following results are the duals of Theorems
2.2(1) and2.4.
Theorem 4.11. LetUandUλbe maximal open sets for any elementλofΛ. IfU⊃λ∈ΛUλ, then there exists an elementλofΛsuch thatU=Uλ.
Theorem 4.12. LetUλ andUγbe maximal open sets for any elementsλofΛandγofΓ. If there exists an elementγofΓsuch thatUλ=Uγ for any elementλofΛ, thenγ∈ΓUγ⊃
λ∈ΛUλ.
References
[1] F. Nakaoka and N. Oda, Some applications of minimal open sets, International Journal of Mathe-matics and Mathematical Sciences 27 (2001), no. 8, 471–476.
[2] , Some properties of maximal open sets, International Journal of Mathematics and Math-ematical Sciences 2003 (2003), no. 21, 1331–1340.
Fumie Nakaoka: Department of Applied Mathematics, Fukuoka University, Nanakuma Jonan-ku, Fukuoka 814-0180, Japan
E-mail address:[email protected]
Nobuyuki Oda: Department of Applied Mathematics, Fukuoka University, Nanakuma Jonan-ku, Fukuoka 814-0180, Japan
