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SOME APPLICATIONS OF MINIMAL OPEN SETS
FUMIE NAKAOKA and NOBUYUKI ODA (Received 16 January 2001)
Abstract.We characterize minimal open sets in topological spaces. We show that any nonempty subset of a minimal open set is pre-open. As an application of a theory of minimal open sets, we obtain a sufficient condition for a locally finite space to be a pre- Hausdorff space.
2000 Mathematics Subject Classification. 54A05, 54D99.
1. Introduction. LetXbe a topological space. We call a nonempty open setUofX a minimal open set when the only open subsets ofUareUand∅.
In this paper, we study fundamental properties of minimal open sets and apply them to obtain some results on pre-open sets (cf. [2]) and pre-Hausdorff spaces.
InSection 2, we characterize minimal open sets, that is, we show that a nonempty open setUis a minimal open set if and only if Cl(U )=Cl(S)for any nonempty subset S ofU. This result implies that any nonempty subsetSof a minimal open setUis a pre-open set.
InSection 3, we study minimal open sets in locally finite spaces. The results of this section are closely related to the work of James [1], and these results will be used in the next scetion.
InSection 4, we apply the theory of minimal open sets to study pre-open sets. Our first main result of this section is a property of the set of all minimal open sets in any nonempty finite open set which is not a minimal open set. This result enables us to prove a generalization ofTheorem 2.5, whenU is a nonempty finite open set, inTheorem 4.4.Theorem 4.5shows that our theory of minimal open set is useful to study pre-open sets.
Finally, we show that some conditions on minimal open sets implies pre-Hausdorff- ness of a space, that is, if any minimal open set of a locally finite spaceX has two elements at least, thenXis a pre-Hausdorff space.
2. Minimal open sets. Let(X, τ)be a topological space.
Definition2.1. A nonempty open setUofXis said to be a minimal open set if and only if any open set which is contained inUis∅orU.
Lemma2.2. (1)LetUbe a minimal open set andWan open set. ThenU∩W= ∅or U⊂W.
(2)LetUandVbe minimal open sets. ThenU∩V= ∅orU=V.
Proof. (1) LetW be an open set such thatU∩W≠∅. SinceU is a minimal open set andU∩W⊂U, we haveU∩W=U. ThereforeU⊂W.
(2) IfU∩V≠∅, then we see thatU⊂V andV⊂Uby (1). ThereforeU=V.
Proposition2.3. LetUbe a minimal open set. Ifxis an element ofU, thenU⊂W for any open neighborhoodWof x.
Proof. LetW be an open neighborhood ofxsuch thatU⊂W. ThenU∩Wis an open set such thatU∩W⊊UandU∩W≠∅. This contradicts our assumption that Uis a minimal open set.
Proposition2.4. LetUbe a minimal open set. Then
U= ∩{W|W is an open neighborhood ofx} (2.1) for any elementxof U.
Proof. ByProposition 2.3and the fact thatUis an open neighborhood ofx, we haveU⊂ ∩{W|W is an open neighborhood ofx} ⊂U. Therefore we have the result.
Theorem2.5. LetU be a nonempty open set. Then the following three conditions are equivalent:
(1)Uis a minimal open set.
(2)U⊂Cl(S)for any nonempty subsetSofU.
(3) Cl(U )=Cl(S)for any nonempty subsetSofU.
Proof. (1)⇒(2). LetS be any nonempty subset of U. ByProposition 2.3, for any elementxofUand any open neighborhoodWofx, we have
S=U∩S⊂W∩S. (2.2)
Then, we haveW∩S≠∅and hencexis an element of Cl(S). It follows thatU⊂Cl(S).
(2)⇒(3). For any nonempty subset S of U, we have Cl(S)⊂Cl(U ). On the other hand, by (2), we see Cl(U )⊂Cl(Cl(S))=Cl(S). Therefore we have Cl(U )=Cl(S)for any nonempty subsetSofU.
(3)⇒(1). Suppose thatUis not a minimal open set. Then there exists a nonempty open setVsuch thatV⊊Uand hence there exists an elementa∈Usuch thata∉V.
Then we have Cl({a})⊂Vc, the complement ofV. It follows that Cl({a})≠Cl(U ).
A subsetMof a space(X, τ)is called apre-openset ifM⊂Int Cl(M). The family of all pre-open sets in(X, τ)will be denoted by PO(X, τ), (cf. [2]).
A space(X, τ)is calledpre-Hausdorffif for eachx,y∈X,x≠ythere exist subsets U,V∈PO(X, τ)such thatx∈U,y∈V, andU∩V= ∅.
Theorem2.6. LetUbe a minimal open set. Then any nonempty subsetS ofUis a pre-open set.
Proof. ByTheorem 2.5(2), we have IntU⊂Int Cl(S). Since U is an open set, we haveS⊂U=Int(U )⊂Int Cl(S).
Theorem2.7. LetUbe a minimal open set andMa nonempty subset ofX. If there exists an open neighborhoodWofMsuch thatW⊂Cl(M∪U ), thenM∪Sis a pre-open set for any nonempty subsetS ofU.
Proof. ByTheorem 2.5(3), we have Cl(M∪S)=Cl(M)∪Cl(S)=Cl(M)∪Cl(U )= Cl(M∪U ). Since W ⊂ Cl(M∪U ) = Cl(M∪S) by assumption, we have Int(W ) ⊂ Int Cl(M∪S). SinceWis an open neighborhood ofM, namelyWis an open set such that M⊂W, we haveM⊂W=Int(W )⊂Int Cl(M∪S). Moreover we have Int(U )⊂Int Cl(M∪ U ), for Int(U )=U⊂Cl(U )⊂Cl(M)∪Cl(U )=Cl(M∪U ). SinceUis an open set, we haveS ⊂U=IntU⊂Int Cl(M∪U )=Int Cl(M∪S). ThereforeM∪S ⊂Int Cl(M∪S).
Corollary2.8. LetUbe a minimal open set andMa nonempty subset ofX. If there exists an open neighborhoodWofMsuch thatW⊂Cl(U ), thenM∪S is a pre-open set for any nonempty subsetSofU.
Proof. By assumption, we haveW⊂Cl(M)∪Cl(U )=Cl(M∪U ). So byTheorem 2.7, we see thatM∪S is a pre-open set.
The condition ofTheorem 2.7, namely W⊂Cl(M∪S), does not necessarily imply the condition ofCorollary 2.8, namelyW⊂Cl(S). We have the following example.
Example2.9. LetX= {a, b, c, d}with topologyθ={∅,{d},{a, b},{a, b, c},{a, b, d}, X}, U= {a, b}and M=W = {d}. ThenW= {d} ⊂Cl({a, b} ∪ {d})=Cl(M∪U )and W= {d} ⊂Cl({a, b})=Cl(U ).
Theorem 2.10. LetU be a minimal open set andx an element of X−U. Then W∩U= ∅orU⊂Wfor any open neighborhoodW of x.
Proof. SinceWis an open set, we have the result byLemma 2.2.
Corollary2.11. LetUbe a minimal open set andxan element ofX−U. Define Ux≡ ∩{W|W is an open neighborhood ofx}. ThenUx∩U= ∅orU⊂Ux.
Proof. IfU⊂Wfor any open neighborhoodWofx, thenU⊂ ∩{W|Wis an open neighborhood ofx}. ThereforeU⊂Ux. Otherwise there exists an open neighborhood Wofxsuch thatW∩U= ∅. Then we haveU∩Ux= ∅.
3. Finite open sets. In this section, we study some properties of minimal open sets in finite open sets and locally finite spaces.
Theorem3.1. LetV be a nonempty finite open set. Then there exists at least one (finite) minimal open setUsuch thatU⊂V.
Proof. IfVis a minimal open set, we may setU=V. IfVis not a minimal open set, then there exists an (finite) open setV1such that∅≠V1⊊V. IfV1is a minimal open set, we may setU=V1. IfV1is not a minimal open set, then there exists an (finite) open setV2such that∅≠V2⊊V1⊊V. Continuing this process, we have a sequence of open sets
V⊋V1⊋V2···⊋Vk⊋···. (3.1)
SinceVis a finite set, this process repeats only finitely. Then, finally we get a minimal open setU=Vnfor some positive integern.
A topological space is said to be a locally finite spaceif each of its elements is contained in a finite open set.
Corollary3.2. LetXbe a locally finite space andV a nonempty open set. Then there exists at least one (finite) minimal open setUsuch thatU⊂V.
Proof. SinceV is a nonempty set, there exists an element x ofV. Since X is a locally finite space, we have a finite open setVxsuch thatx∈Vx. SinceV∩Vxis a finite open set, we get a minimal open setUsuch thatU⊂V∩Vx⊂VbyTheorem 3.1.
Theorem3.3. LetVλ be an open set for anyλ∈ΛandW a nonempty finite open set. ThenW∩(∩λ∈ΛVλ)is a finite open set.
Proof. We see that there exists an integer n such that W∩(∩λ∈ΛVλ) = W∩ (∩ni=1Vλi)and hence we have the result.
Theorem3.4. LetVλbe an open set for anyλ∈ΛandWµa nonempty finite open set for anyµ∈ᏹ. LetS= ∪µ∈ᏹWµ. ThenS∩(∩λ∈ΛVλ)is an open set.
Proof. SinceWµis a finite open set, byTheorem 3.3, we haveWµ∩(∩λ∈ΛVλ)is a finite open set for anyµ∈ᏹ. Since
S∩
∩λ∈ΛVλ
=
∪µ∈ᏹWµ
∩
∩λ∈ΛVλ
= ∪µ∈ᏹ Wµ∩
∩λ∈ΛVλ
, (3.2)
we have the result.
Corollary3.5(see [1]). Any locally finite space is an Alexandroff space.
4. Applications. LetUbe a nonempty finite open set. We see, byLemma 2.2and Corollary 3.2, that there exists a positive integerksuch that{U1, U2, . . . , Uk}is the set of all minimal open sets inU. Then it satisfies the following two conditions:
(a)Ui∩Uj= ∅for anyi,jwith 1≤i,j≤k, andi≠j.
(b) IfUis a minimal open set in U, then there existsiwith 1≤i≤k such that U=Ui.
Theorem 4.1. LetU be a nonempty finite open set which is not a minimal open set. Let{U1, U2, . . . , Un}be the set of all minimal open sets inUandx an element of U−(U1∪U2∪ ··· ∪Un). DefineUx≡ ∩{W|W is an open neighborhood ofx}. Then there exists a positive integeriof{1, . . . , n}such thatUi⊂Ux.
Proof. Assume thatUi⊂Uxfor any positive integeriof{1, . . . , n}. Then we have Ui∩Ux= ∅for any minimal open setUiinUbyCorollary 2.11. SinceUxis a nonempty finite open set byTheorem 3.3, there exists a minimal open setUsuch thatU⊂Uxby Theorem 3.1. SinceU⊂Ux⊂U, we haveUis a minimal open set inU. By assumption, we haveUi∩U⊂Ui∩Ux= ∅for any minimal open setUi. ThereforeU≠Uifor any positive integeriof{1,2, . . . , n}. This contradicts our assumption.
Proposition4.2. LetUbe a nonempty finite open set which is not a minimal open set. Let{U1, U2, . . . , Un}be the set of all minimal open sets inUandx an element of U−(U1∪U2∪ ··· ∪Un). Then there exists a positive integeriof{1, . . . , n}such that Ui⊂Wxfor any open neighborhoodWxofx.
Proof. SinceWx⊃ ∩{W|Wis an open neighborhood ofx}, we have the result by Theorem 4.1.
Theorem 4.3. LetU be a nonempty finite open set which is not a minimal open set. Let{U1, U2, . . . , Un}be the set of all minimal open sets inUandx an element of U−(U1∪U2∪···∪Un). Then there exists a positive integeriof{1, . . . , n}such thatx is an element ofCl(Ui).
Proof. ByProposition 4.2, there exists a positive integeriof{1, . . . , n}such that Ui⊂W for any open neighborhoodW ofx. ThereforeUi∩W⊃Ui∩Ui≠∅for any open neighborhoodWofx. Therefore we have the result.
The following result is a generalization ofTheorem 2.5, whenUis a nonempty finite open set.
Theorem4.4. LetUbe a nonempty finite open set andUia minimal open set inU for eachi∈ {1,2, . . . , n}. Then the following three conditions are equivalent:
(1){U1, U2, . . . , Un}is the set of all minimal open sets inU.
(2)U⊂Cl(S1∪S2∪···∪Sn)for any nonempty subsetsSiofUifori∈ {1,2, . . . , n}. (3) Cl(U )=Cl(S1∪S2∪···∪Sn)for any nonempty subsetsSiofUifori∈{1,2, . . . , n}. Proof. (1)⇒(2). IfUis a minimal open set, then this is the result ofTheorem 2.5(2).
OtherwiseUis not a minimal open set. Ifxis any element ofU−(U1∪U2∪···∪Un), we havex∈Cl(U1)∪Cl(U2)∪···∪Cl(Un)byTheorem 4.3. Therefore
U⊂Cl U1
∪Cl U2
∪···∪Cl Un
=Cl S1
∪Cl S2
∪···∪Cl Sn
=Cl
S1∪S2∪···∪Sn
(4.1)
byTheorem 2.5(3).
(2)⇒(3). For any nonempty subsetSiofUiwithi∈ {1,2, . . . , n}, we have Cl(S1∪S2∪
···∪Sn)⊂Cl(U ). On the other hand, by (2), we see Cl(U )⊂Cl
Cl
S1∪S2∪···∪Sn
=Cl
S1∪S2∪···∪Sn
. (4.2)
Therefore we have Cl(U )=Cl(S1∪S2∪···∪Sn)for any nonempty subsetSiofUiwith i∈ {1,2, . . . , n}.
(3)⇒(1). Suppose thatV is a minimal open set inUandV≠Uifori∈ {1,2, . . . , n}. Then we haveV∩Cl(Ui)= ∅for eachi∈ {1,2, . . . , n}. It follows that any element ofV is not contained in Cl(U1∪U2∪ ··· ∪Un). This contradicts the condition (3) because V⊂U⊂Cl(U )=Cl(S1∪S2∪···∪Sn).
LetUbe a nonempty finite open set,{U1, U2, . . . , Un}the set of all minimal open sets inUandxian element ofUifor eachi∈ {1,2, . . . , n}. Then we see that the set{x1, x2, . . . , xn}is a pre-open set byTheorem 4.4. Moreover, we have the following result.
Theorem4.5. LetUbe a nonempty finite open set and{U1, U2, . . . , Un}the set of all minimal open sets inU. LetS be any subset ofU−(U1∪U2∪ ··· ∪Un)andSibe any nonempty subset ofUifor eachi∈{1,2, . . . , n}. ThenS∪S1∪S2···∪Snis a pre-open set.
Proof. ByTheorem 4.4(2), we have U⊂Cl
S1∪S2···∪Sn
⊂Cl
S∪S1∪S2···∪Sn
. (4.3)
SinceUis an open set, then we have
S∪S1∪S2···∪Sn⊂U=Int(U )⊂Int Cl
S∪S1∪S2···∪Sn
. (4.4)
Then we have the result.
Theorem4.6. LetXbe a locally finite space. If any minimal open set ofXhas two elements at least, thenXis a pre-Hausdorff space.
Proof. Letx,ybe elements ofXsuch thatx≠y. SinceXis a locally finite space, there exists finite open setsUandVsuch thatx∈Uandy∈V. ByTheorem 3.1, there exists the set{U1, U2, . . . , Un}of all minimal open sets inUand the set{V1, V2, . . . , Vm} of all minimal open sets inV.
Case1. If there existsiof{1,2, . . . , n}andj of{1,2, . . . , m}such thatx∈Uiand y∈Vj, then, byTheorem 2.6,{x}and{y}are disjoint pre-open sets which contains xandy, respectively.
Case2. If there existsiof{1,2, . . . , n}such thatx∈Ui andy∈Vj for anyj of {1,2, . . . , m}, then we find an elementyjofVjfor eachj such that{x}and{y, y1, y2, . . . , ym}are pre-open sets and{x} ∩ {y, y1, y2, . . . , yn} = ∅by Theorems2.6,4.5 and the assumption.
Case3. Ifx∈Uifor anyiof{1,2, . . . , n}andy∈Vjfor anyjof{1,2, . . . , m}, then we find elementsxiofUi andyj ofVjfor eachi,jsuch that{x, x1, x2, . . . , xn}and {y, y1, y2, . . . , ym}are pre-open sets and{x, x1, x2, . . . , xn} ∩ {y, y1, y2, . . . , ym} = ∅ byTheorem 4.5and the assumption. We remark that we use the assumption that any minimal open set ofXhas at least two elements for the caseUi=Vjfor someiand jin the argument of cases (2) and (3).
ThereforeXis a pre-Hausdorff space.
References
[1] I. M. James,Alexandroff spaces, Rend. Circ. Mat. Palermo (2) Suppl. (1992), no. 29, 475–
481, International Meeting on Topology in Italy (Italian) (Lecce, 1990/Otranto, 1990).
MR 94g:54020. Zbl 793.54006.
[2] A. S. Mashhour, M. E. Abd El-Monsef, and S. N. El-Deep,On precontinuous and weak precon- tinuous mappings, Proc. Math. Phys. Soc. Egypt (1982), no. 53, 47–53.MR 87c:54002.
Zbl 571.54011.
Fumie Nakaoka: Department of Applied Mathematics, Fukuoka University, Nanakuma Jonan-ku, Fukuoka814-0180, Japan
E-mail address:[email protected]
Nobuyuki Oda: Department of Applied Mathematics, Fukuoka University, Nanakuma Jonan-ku, Fukuoka814-0180, Japan
E-mail address:[email protected]
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