C om mun.Fac.Sci.U niv.A nk.Series A 1 Volum e 67, N umb er 1, Pages 141–146 (2018) D O I: 10.1501/C om mua1_ 0000000837 ISSN 1303–5991
http://com munications.science.ankara.edu.tr/index.php?series= A 1
A NOTE ON TOPOLOGIES GENERATED BY m-STRUCTURES
AND !-TOPOLOGIES
AHMAD AL-OMARI AND TAKASHI NOIRI
Abstract. Let (resp. SO(X; )) be the family of all -open (resp. semi-open) sets in a topological space (X; ). The topology is constructed in [10] as follows: = T (SO(X)) = fU X : U \ S 2 SO(X; ) for every S 2 SO(X; )g. By the same method, we construct topologies T (mX)and T (!mX)for m-structrues mXand !mXde…ned in [11], respectively, and show that !T (mX) T (!mX). Furthermore, in [2], a topology M is constructed by using an M -space (X; M) with an ideal I. In this note, we de…ne !M-open sets on (X; M) and show that the family !M of all !M-open sets is a topology for X and !(M ) = (!M) = (!M) :
1. Introduction
In 1982, Hdeib [6] introduced and investigated the notions of !-closed sets and !-closed mappings. Al-Zoubi and Al-Nashef [3] investigated several properties of the topology of !-open sets. Recently, Noiri and Popa [11] have introduced the notion of !m-open sets in an m-space and, by utilizing !m-open sets, obtained several properties of m-Lindel•of spaces. Let (resp. SO(X; )) be the family of all -open (resp. semi-open) sets of a topological space (X; ). Then SO(X; ) is not a topology but the topology is constructed in [10] as follows: fU X : U \ S 2 SO(X; ) for every S 2 SO(X; )g = . In this note, by the same method we construct some topologies from an m-structure and the family of !m-open sets and investigate their relations.
On the other hand, Al-Omari and Noiri [2] constructed the topology M from an M -space (X; M) with an ideal I. In this note, we de…ne the notion of !M-open sets in (X; M) and show that the family !M of all !M-open sets is a topology for X and also !(M ) = (!M) .
Received by the editors: December 29, 2016, Accepted: April 17, 2017. 2010 Mathematics Subject Classi…cation. Primary 54A05; Secondary 54C10. Key words and phrases. m-structure, !-open, !m-open, !-topology, -topology.
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2. Preliminaries
Let (X; ) be a topological space and A a subset of X. The closure of A and the interior of A are denoted by Cl(A) and Int(A), respectively. We recall some de…nitions and theorems used in this note.
De…nition 1. Let (X; ) be a topological space. A subset A of X is said to be !-open [6] if for each x 2 A there exists U 2 containing x such that U n A is a countable set.
The family of all !-open sets in (X; ) is denoted by ! .
Lemma 1. (Al-Zoubi and Al-Nashef [3]). Let (X; ) be a topological space. Then ! is a topology and it is strictly …ner than .
De…nition 2. Let X be a nonempty set and P (X) the power set of X. A subfam-ily mX of P (X) is called an m-structure on X [11] if mX satis…es the following
properties:
(1) ; 2 mX and X 2 mX,
(2) The arbitrary union of the sets belonging to mX belongs to mX.
By (X; mX), we denote a set X with an m-structure mX and call it an m-space.
Each member of mX is said to be mX-open and the complement of an mX-open
set is said to be mX-closed.
De…nition 3. Let (X; mX) be an m-space. A subset A of X is said to be !mX
-open [11] if for each x 2 A, there exists Ux2 mX containing x such that Uxn A is
a countable set. The complement of an !mX-open set is said to be !mX-closed.
The family of all !mX-open sets in (X; mX) is denoted by !mX.
Remark 1. Let (X; ) be a topological space and mX an m-structure on X. If
mX, then the following relations hold. We can observe that the implications in
the diagram below are not reversible.
open? ! mX-open ? ? y ? ? ? y !-open ! !mX-open
Lemma 2. (Noiri and Popa [11]). Let (X; mX) be an m-space and A a subset of
X. Then the following properties hold:
(1) A is !mX-open if and only if for each x 2 A, there exists Ux 2 mX
containing x and a countable subset Cx of X such that Uxn Cx A,
(2) The family !mX is an m-structure on X and !mX is a topology if mX is
a topology,
De…nition 4. Let (X; ) be a topological space. A subset A of X is said to be (1) -open [10] if A Int(Cl(Int(A))),
(2) semi-open [8] if A Cl(Int(A)), (3) preopen [9] if A Int(Cl(A)),
(4) b-open [4] if A Int(Cl(A)) [ Cl(Int(A)), (5) -open [1] if A Cl(Int(Cl(A))).
The family of all -open (resp. semi-open, preopen, b-open, -open) sets in (X; ) is denoted by (resp. SO(X), PO(X), BO(X), (X)).
De…nition 5. For an m-space (X; mX), we de…ne T (mX) as follows:
T (mX) = fU X : U \ MX 2 mX for every MX2 mXg.
Remark 2. Let (X; ) be a topological space and mX = PO(X) (resp. BO(X),
(X)). Then T (mX) is denoted by T (resp. Tb, T ) and the following properties
are known:
(1) T (SO(X)) = [10], (2) T = T [5], and (3) T = Tb [4].
3. Topologies generated by mX and !mX
Theorem 1. Let (X; mX) be an m-space. Then T (mX) is a topology for X such
that T (mX) mX.
Proof. (1) It is obvious that ;; X 2 T (mX).
(2) Let V 2 T (mX) for each 2 . Let A be an arbitrary element of mX. For
each 2 , V \ A 2 mX and f[V : 2 g \ A = [fV \ A : 2 g 2 mX by
De…nition 2 (2). Therefore, we have [ 2 V 2 T (mX).
(3) Let V1; V22 T (mX). For any A 2 mX, V2\ A 2 mX and (V1\ V2) \ A =
V1\ (V2\ A) 2 mX. Therefore, we obtain V1\ V22 T (mX).
Furthermore, for any V 2 T (mX), V = V \ X 2 mX and hence T (mX)
mX.
The following corollary is results established by Nåstad [10], Ganster and Andrijevia [5] and Andrijevia [4].
Corollary 1. Let (X; ) be a topological space. Then the families SO(X), PO(X), BO(X), (X) are m-structures on X. Therefore, , T , Tb and T are topologies
for X.
Theorem 2. Let (X; mX) be an m-space. Then !T (mX) T (!mX).
Proof. Suppose that A 2 !T (mX). To obtain that A 2 T (!mX), we show that
A \ B 2 !mX for every B 2 !mX. For each x 2 A \ B; x 2 A 2 !T (mX) and
by Lemma 2, there exist Ux 2 T (mX) containing x and a countable set Cx such
that Uxn Cx A. On the other hand, since x 2 B 2 !mX, there exist Vx 2 mX
A \ B (Uxn Cx) \ (Vxn Dx) = Ux\ (X n Cx)) \ (Vx\ (X n Dx)) = (Ux\ Vx) \
[(X n Cx) \ (X n Dx)] = (Ux\ Vx) \ [X n (Cx[ Dx)] = (Ux\ Vx) n (Cx[ Dx).
Since Cxand Dxare countable, Cx[ Dx is a countable set. Since Ux2 T (mX)
and Vx2 mX, Ux\ Vx 2 mX and x 2 Ux\ Vx. Therefore, by Lemma 2, A \ B 2
!mX. This shows that A 2 T (!mX). Therefore, !T (mX) T (!mX).
Remark 3. By Lemma 2 and Theorems 1 and 2, we obtain the following diagram:
mX ! !mX x ? ? ? x ? T (!mx X) ? T (mX) ! !T (mX)
QUESTION: Is the converse implication of Theorem 2 true ?
Corollary 2. Let (X; ) be a topological space. Then ! T (!SO(X)). 4. Topologies generated by M -spaces with ideals
De…nition 6. Let X be a nonempty set and P(X) the power set of X. A subfamily M of P(X) is called an M-structure on X [2] if M satis…es the following properties:
(1) M contains ; and X,
(2) M is closed under the …nite intersection.
By (X; M), we denote a set X with an M-structure M and call it an M-space. Each member of M is said to be M-open and the complement of an M-open set is said to be M -closed.
De…nition 7. Let (X; M ) be an M -space. A subset A of X is said to be !M -open if for each x 2 A, there exists Ux2 M containing x such that Uxn A is a countable
set. The complement of an !M -open set is said to be !M -closed. The family of all !M -open sets in (X; M) is denoted by !M.
Lemma 3. Let (X; M) be an M-space and A a subset of X. Then A is !M-open if and only if for each x 2 A, there exists Ux 2 M containing x and a countable
subset Cx of X such that Uxn Cx A.
Proof. Necessity. Let A be !M -open and x 2 A. Then there exists Ux 2 M
containing x such that Uxn A is a countable set. Let Cx= Uxn A. Then we have
Uxn Cx A.
Let x 2 A. Then there exists Ux2 M containing x and a countable set Cxsuch
that Uxn Cx A. Therefore, Uxn A Cx and Uxn A is a countable set. Hence A
is !M -open.
(1) The family !M is a topology for X,
(2) M !M and !(!M) = !M.
Proof. (1) (i) It is obvious that ;; X 2 !M.
(ii) Let A; B 2 !M and x 2 A \ B. Then, by Lemma 3, there exist U; V 2 M and countable sets C; D such that x 2 U and U n C A and x 2 V and V n D B. Therefore, x 2 U \ V 2 M, C [ D is countable and we have
(U \V )n(C [D) = (U \V )\[(X nC)\(X nD)] = [U \(X nC)]\[V \(X nD)] = (U n C) \ (V n D) A \ B.
This shows that A \ B 2 !M.
(iii) Let fA : 2 g be any subfamily of !M. Then for each x 2 [ 2 A ,
there exists (x) 2 such that x 2 A (x). Since A (x)2 !M, there exists Ux2 M
containing x such that Ux n A (x) is a countable set. Since Uxn ([ 2 A )
Uxn A (x), Uxn ([ 2 A ) is a countable set. Therefore, [ 2 A 2 !M. This
shows that !M is a topology.
(2) Since every M -open set is !M -open, M !M. Therefore, by (1) we have !M !(!M). Let A 2 !(!M). By Lemma 3, for each x 2 A, there exists Ux2 !M containing x and a countable set Cxsuch that UxnCx A. Furthermore,
by Lemma 3, there exists Vx2 M containing x and a countable set Dx such that
Vxn Dx Ux. Therefore, we have Vxn (Cx[ Dx) = (Vxn Dx) n Cx Uxn Cx A.
Since Cx[Dxis a countable set, we obtain that A 2 !M. Therefore, !(!M) !M
and hence !(!M) = !M.
A subfamily I of P(X) is called an ideal [7] if it satis…es the following properties: (1) A 2 I and B A imply that B 2 I;
(2) A 2 I and B 2 I imply that A [ B 2 I.
An M -space (X; M) with an ideal I is called an ideal M-space and is denoted by (X; M; I) [2]. In [2], for a subset A of X the M-local function of A is de…ned as follows:
A (I; M) = fx 2 X : A \ U =2 I for every U 2 M(x)g,
where M(x) = fU 2 M : x 2 Ug. In case there exists no confusion A (I; M) is brie‡y denoted by A . By Theorem 4.2 of [2], it is shown that Cl (A) = A [ A is a Kuratowski closure operator. The topology generated by Cl is denoted by M , that is, M = fU X : Cl (X n U) = X n Ug.
Lemma 4. (Al-Omari and Noiri [2]) Let (X; M; I) be an ideal M-space. Then (M; I) = fV n I : V 2 M; I 2 Ig is a basis for M .
Theorem 4. For any ideal M -space (X; M; I), !(M ) = (!M) .
Proof. First, we show that !(M ) (!M) . Let A 2 (!M) and x 2 A. Then, by Lemma 4, there exist V 2 !M and I 2 I such that x 2 V n I A. Since x 2 V 2 !M, there exist Gx2 M and a countable set Cxsuch that x 2 Gxn Cx V .
Therefore, x 2 Gxn I 2 M and (Gxn I) n Cx= (Gxn Cx) n I V n I A. This
Next, we show that !(M ) (!M) . Let A 2 !(M ) and x 2 A. Then there exist Vx2 M and a countable set Cx such that x 2 Vxn Cx A. Since Vx2 M ,
by Lemma 4, there exist Gx 2 M and I 2 I such that x 2 Gxn I Vx. Then
x 2 Gxn Cx 2 !M and (Gxn Cx) n I = (Gxn I) n Cx Vxn Cx A. This
shows that A 2 (!M) . Therefore, !(M ) (!M) . Consequently, we obtain that !(M ) = (!M) .
In an ideal topological space (X; ; I), the topology generated by the local func-tion is denoted by . It is known in [7] that = if I is the nowhere dense ideal. Thus, we have the following corollaries:
Corollary 3. For any ideal M -space (X; M; I), !(M ) = (!M) = (!M) . Corollary 4. Let (X; ; I) be an ideal topological space. Then !( ) = (! ) . Corollary 5. Let (X; ; I) be an ideal topological space. If I is the nowhere dense ideal, then ! = (! ) .
acknowledgement
The authors wish to thank the referees for useful comments and suggestions. References
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Current address : Ahmad Al-Omari: Al al-Bayt University, Faculty of Sciences, Department of Mathematics, P.O. Box 130095, Mafraq 25113, Jordan
E-mail address : omarimutah1@yahoo.com
Current address : Takashi Noiri: 2949-1 Shiokita-cho, Hinagu, Yatsushiro-shi, Kumamoto-ken, 869-5142 Japan
