- Email: [email protected]
Transfer closed and transfer open multimaps in minimal spaces
Chaos, Solitons and Fractals 40 (2009) 1162–1168 www.elsevier.com/locate/chaos
Transfer closed and transfer open multimaps in minimal spaces M. Alimohammady *, M. Roohi, M.R. Delavar Department of Mathematics, University of Mazandaran, Babolsar 47416-1468, Iran Accepted 29 August 2007
Abstract This paper is devoted to introduce the concepts of transfer closed and transfer open multimaps in minimal spaces. Also, some characterizations of them are considered. Further, the notion of minimal local intersection property will be introduced and characterized. Moreover, some maximal element theorems via minimal transfer closed multimaps and minimal local intersection property are given. 2007 Elsevier Ltd. All rights reserved.
1. Introduction and preliminaries It is well known that topological concepts have many applications in modern physics. For example, the topology of quantum spacetime is shadowed closely by the Mobius geometry of quasi-Fuschian and Kleinian groups and that is the cause behind the phenomena of high-energy particle physics [11]. In fact, considering the spacetime as the product of two topologies, the topology of space and that of the spacetime will open the way for new line of research in the field of quantum gravity initiated by Witten and El-Naschie. For the relation between Wild Topology, Hyperbolic Geometry and Fusion Algebra with Coupling constants of the standard model and quantum gravity and close connection between e1 theory and the topological theory of four manifolds, we refer to [12,13] and for the relation between topological concepts and geometrical properties of the e1 spacetime to [14]. For another example, constructing a topology via a relation on a real-life data will help in mathematizing many fields. In fact, if X is a collection of symptoms and diseases in a certain region and R is a binary relation on X given by an expert the topology on X generated by R is a knowledge base for X, indication of symptoms for a fixed disease can be seen through the topology [7]. Since topology has very important applications in applied sciences, so studying of minimal structure as a generalization of topology is important from this point of view. Minimal structures may have very important applications in quantum particles physics, particularly in connection with string theory and e1 theory [6,8–10]. The work presented
*
Corresponding author. E-mail addresses: [email protected] (M. Alimohammady), [email protected] (M. Roohi), [email protected] (M.R. Delavar). 0960-0779/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.08.071
M. Alimohammady et al. / Chaos, Solitons and Fractals 40 (2009) 1162–1168
1163
in our paper, not initiate new classes with respect to topology but in view of minimal structures. In this paper after introducing transfer closed and transfer open multimaps via minimal space, some characterizations of them are given. Moreover, the concept of minimal local intersection property will be introduced and some its characterization will be investigated. Finally, some maximal element theorems via minimal transfer closed multimaps and minimal local intersection property are considered. The concepts of minimal structure and minimal spaces, as a generalization of topology and topological spaces were introduced in [21]. For easy understanding of the material incorporated in this paper we recall some basic definitions. For details on the following notions we refer to [1–4,21,22] and [24] and references therein. A family M # PðX Þ is said to be a minimal structure on X if ;, X 2 M. In this case ðX ; MÞ is called a minimal space. For some examples in this setting see [21,24]. In a minimal space ðX ; MÞ, A 2 PðX Þ is said to be an m-open set if A 2 M and also B 2 PðX Þ is an m-closed set if Bc 2 M. For any x 2 X, N(x) is said to be a minimal neighborhood of x, if for any S z 2 N(x) there is an m-open subset G N(x) such that z 2 G . Set m IntðAÞ ¼ fU : U # A; U 2 Mg and z z T m ClðAÞ ¼ fF : A # F ; F c 2 Mg. Proposition 1.1. [21] For any two sets A and B, (a) (b) (c) (d) (e) (f) (g)
m-Int(A) A and m-Int(A) = A if A is an m-open set, A m-Cl(A) and A = m-Cl(A) if A is an m-closed set, m-Int(A) m-Int(B) and m- Cl(A) m-Cl(B) if A B, m-Int(A \ B) = (m-Int(A)) \ (m-Int(B)) and (m-Int(A)) [ (m-Int(B)) m-Int(A [ B), m-Cl(A [ B) = (m-Cl(A)) [ (m-Cl(B)) and m -Cl(A \ B) (m-Cl(A)) \ (m-Cl(B)), m-Int (m-Int(A)) = m-Int(A) and m–Cl(m–Cl(B)) = m-Cl(B), (m-Cl(A))c = m-Int(Ac) and (m-Int(A))c = m-Cl(Ac).
2. Main Results Suppose X and Y are two minimal spaces. A multimap F: X ( Y is a function from a set X into the power set of Y; that is, a function with the values F(x) Y for all x 2 X. Given A X, set [ F ðxÞ: F ðAÞ ¼ x2A
We say that a multimap F: X ( Y has a maximal element if F(x0) = ;, for some x0 2 X. For a multimap F: X ( Y, the multimaps Fc, m–Cl(F), and m–Int(F) from X to Y are defined by Fc(x) = {y 2 Y: y R F(x)}, (m–Cl(F))(x) = m–Cl(F(x)) and (m–Int(F))(x) = m–Int(F(x)), respectively. Also multimaps F and F* from Y to X are defined by F(y) = {x 2 X:y 2 F(x)} and F*(y) = {x 2 X:y R F(x)}, respectively. Some properties of these multimaps and their relations can be found in [17,19]. Lemma 2.1. [19] Suppose F, G: X ( Y are two multimaps. Then (a) (b) (c) (d) (e) (f)
for each x 2 X, F(x) G(x) if and only if G*(y) F*(y) for each y 2 Y, y R F(x) if and only if x 2 F*(y), for each x 2 X, (F*)*(x) = F(x), T for each x 2 X, F(x) 5 ; if and only if F*(y) = ;, y2A for each y 2 Y, (Fc)*(y) = F(y), c * for each y 2 Y, (F ) (y) = F (y).
Proposition 2.1. Suppose F: X ( Y is a multimap. Then (a) (b) (c) (d)
(m–Cl(F))c = m–Int(Fc), (m–Cl(F))* = (m– Int(Fc)), (m–Cl(F*))c = m–Int(F), (m–Cl(F*))* = (m–Int(F)).
Proof. It is an immediate consequence of Proposition 1.1 and Lemma 2.1.
h
1164
M. Alimohammady et al. / Chaos, Solitons and Fractals 40 (2009) 1162–1168
Definition 2.1. Suppose X and Y are two minimal spaces. A multimap F: X ( Y is said to be (a) minimal transfer open if for each x 2 X and y 2 F(x), there exists x0 2 X for which y 2 m–Int(F(x0)), (b) minimal transfer closed if for any x 2 X and y R F(x), there exists x0 2 X for which y R m–Cl(F(x0)). Theorem 2.1. Suppose X and Y are two minimal spaces. The following are equivalent. (a) The S F: X ( Y is minimal transfer open, S multimap F(x) = m–Int(F(x)), (b) x2X x2A T T c F (x) = m–Cl(Fc(x)), (c) x2X x2X T T c F (x) = (m-Int(F(x)))c. (d) x2X
x2X
Proof. The implications ðbÞ $ ðcÞ, ðcÞ $ ðdÞ and ðbÞ ! ðaÞ are straightforward. OnlySwe must prove the S implication ðaÞ ! ðbÞ. For this, assume (a) is true. It follows from part (a) of Proposition 1.1 that m–Int(F(x)) F(x). On the x2X x2X S other hand, for y 2 F(x), there is x1 2 X such that y 2 F(x1). Then from (a) there is x0 2 X for which y 2 m–Int(F(x0)) S Sx2X and so F(x) m–Int(F(x)); i.e., (b) satisfies, which it completes the proof. h x2X
x2X
Theorem 2.2. Suppose X and Y are two minimal spaces. A multimap F: X ( Y is minimal transfer open if and only if Fc is minimal transfer closed. Proof. Suppose F is minimal transfer open. Consider x 2 X and y 2 Y in which y R Fc(x); i.e., y 2 F(x). By the assumption, there is x0 2 X for which y 2 m–Int(F(x0)). Therefore, y R m–Cl(Fc(x0)) and so Fc is minimal transfer closed. Conversely, suppose Fc is minimal transfer closed. For any x 2 X, if y 2 F(x) then y R Fc(x). From the assumption there is x0 2 X such that y R m–Cl(Fc(x0)). Therefore, y 2 m–Int(F(x0)). h Theorem 2.3. Suppose X and Y are two minimal spaces. The following are equivalent. (a) The T F:X ( Y is minimal transfer closed, T multimap F(x) = m–Cl(F(x)), (b) x2X x2X S S c F (x) = m–Int(Fc(x)), (c) x2X x2X S S c F (x) = (m–Cl(F(x)))c. (d) x2X
x2X
Proof. It is an immediate consequence of Theorems 2.1 and 2.2.
h
Remark 2.1. Theorems 2.2 and 2.3 are extended versions of Lemma 2.1 in [5]. Definition 2.2. Suppose X and Y are two minimal spaces. A multimap F:X ( Y is said to have minimal local intersection property if for each x 2 X with F(x) 5 ;, there is minimal neighborhood N(x) of x for which \ F ðyÞ–;: y2mIntðN ðxÞÞ
Proposition 2.2. Suppose X and Y are two minimal spaces. Then F: X ( Y has the minimal local intersection property if and only if for each x 2 F(y), there is a minimal neighborhood N(x) of x and y1 2 Y for which m– Int(N(x)) F(y1). Proof. Suppose F has the minimal local intersection property and that x 2 F(y). Therefore, y 2 F(x); i.e. F(x) 5 ; and so by the assumption there is a minimal neighborhood N(x) of x for which \ F ðyÞ–;: y2mIntðN ðxÞÞ
M. Alimohammady et al. / Chaos, Solitons and Fractals 40 (2009) 1162–1168
1165
Then there is y1 2 Y with y1 2 F(z) for all z 2 m–Int(N(x)). Consequently, m-Int(N(x)) F(y1). Conversely, suppose F(x) 5 ;, there is y 2 Y for which x 2 F(y). There exits a minimal neighborhood N(x) of x and y1 2 Y in which m–Int(N(x)) F(y1). Therefore, z 2 F(y1) for all z 2 m–Int(N(x)) which completes the proof. h Theorem 2.4. Suppose X and Y are two minimal spaces. Then F: X ( Y has the minimal local intersection property if and only if F: Y ( X is minimal transfer open. Proof. Suppose F has the minimal local intersection property and x 2
S
F(y). There is y0 2 Y in which x 2 F(y0) and
y2Y
so y0 2 F(x). According to Proposition 2.2 there is minimal neighborhood N(x) of x and y1 2 Y in which mInt(N(x)) F(y1). Therefore, x 2 m IntðN ðxÞÞ # m IntðF ðy 1 ÞÞ #
[
m IntðF ðyÞÞ;
y2Y
i.e., F is minimal transfer open. Conversely, suppose that F is minimal transfer open. Consider an arbitrary element x 2 X with F(x) 5 ;. There is y1 2 F(x) and so x 2 F(y1). Since F is minimal transfer open, so Theorem 2.1 implies that, there isT an element y0 2 Y such that x 2 m–Int(F(y0)). Now, if z 2 m–Int(F(y0)) then z 2 F(y0) and so y0 2 F(z). Hence, F ðzÞ–;. h z2mIntðF ðy 0 ÞÞ
Corollary 2.1. Suppose X and Y are two minimal spaces. Then (a) F: X ( Y has the minimal local intersection property if and only if F*:Y ( X is minimal transfer closed, (b) F: X ( Y is minimal transfer closed if and only if F*:Y ( X has the minimal local intersection property. Proof. The first assertion follows from Theorem 2.4 and part (f) of Lemma 2.1. Also, the second part immediately follows from (c) in Lemma 2.1 and (a) in this corollary. h Remark 2.2. It should be noticed that the origin versions of Proposition 2.2, Theorem 2.4 and Corollary 2.1 go back to Lan and Wu [19]. Proposition 2.3. Suppose X and Y are two minimal spaces. Then the following statements for a multimap F: X ( Y are equivalent. (a) F: Y ( X is minimal transfer open and F(x) 5 ; for all x 2 X, (b) F: X ( Y has the minimal local intersection property and F(x) 5 ; for all x 2 X, (c) X = ¨{m–Int(F(y)): y 2 Y}. Proof. That (a) and (b) are equivalent follows from Theorem 2.4. Suppose (b) satisfies and x 2 X is arbitrary. Since F(x) 5 ; so there is a minimal neighborhood N(x) of x for which \ F ðyÞ–;: y2mIntðN ðxÞÞ
There exists y0 2 Y such that y0 2 F(z) for all z 2 m–Int(N(x)); i.e., z 2 F(y0) for all z 2 m–Int(N(x)). Therefore, x 2 m IntðN ðxÞÞ # m IntðF ðy 0 ÞÞ; and so x 2 m–Int(F(y0)). S S For implication ðcÞ ) ðaÞ, suppose X = ¨{m–Int(F(y)): y 2 Y}. We must show that F (y) = m–Int(F(y)). y2Y y2Y One direction is straightforward, and other follows from the fact that [ [ F ðyÞ # X ¼ m IntðF ðyÞÞ: y2Y
y2Y
Also for each x 2 X, there is y0 2 Y for which x 2 m-Int(F(y0)) and so x 2 F(y0). Then y0 2 F(x); i.e., F(x) 5 ; for all x 2 X. h
1166
M. Alimohammady et al. / Chaos, Solitons and Fractals 40 (2009) 1162–1168
Theorem 2.5. Suppose X and Y are two minimal spaces. For a multimap F: X ( Y the following statements are equivalent. (a) FThas the minimalTlocal intersection property and F(x) 5 ; for all x 2 X, m-Cl(F*(y)) = F*(y) = ;, (b) y2Y
y2Y
(c) (m-Cl(F*))*(x) 5 ; for each x 2 X, S (d) X = m–Int(F(y)). y2Y
Proof. ðaÞ ) ðbÞ follows from Corollary 2.1, (a) and (b) in Theorem 2.3 and (d) in Lemma 2.1. Suppose T m–Cl(F*(y)) = ;, so for any x 2 X there exists y 2 Y such that x R m–Cl(F*(y)). This mean that y 2 (m–Cl(F*))*(x). y2Y
Therefore for any x 2 X, (m–Cl(F*))*(x) 5 ; and so ðbÞ ) ðcÞ is proved. For ðcÞ ) ðdÞ, suppose that (c) is true. So *))*(x). Part (d) in Proposition 2.1, implies that y 2 (m–Int(F))(x) for any x 2 X there is y 2 Y in which y 2 (m–Cl(F S and therefore x 2 (m–Int(F )(y)). Hence X = m–Int(F)(y) and so the implication ðcÞ ) ðdÞ is proved. Finally, supy2Y T T S pose that X = m–Int(F)(y), so (m-Int(F))c(y) = ;. Condition (c) of Proposition 2.1 shows that F*(y) = ; and y2Y
y2Y
y2Y
from (d) in Lemma 2.1, F(x) 5 ; for any x 2 X. Also from Theorem 2.3 and Corollary 2.1, F has the minimal local intersection property. h Theorem 2.6. Suppose F, G: X ( Y are two multimaps satisfying (a) for each x 2 X, F(x) G(x), (b) F has the minimal local intersection property. Then one of the following statements hold. (1) G has the minimal local intersection property, (2) F has a maximal element. Proof. Suppose x 2 X with G(x) 5 ;. If F has no maximal element, then condition (b) implies that there is a minimal neighborhood N(x) of x for which ˙{F(z): z 2 m–Int(N(x))} 5 ; and hence by (a) we have ˙{G(z): z 2 m– Int(N(x))} 5 ;; i.e., (1) holds. h Theorem 2.7. The following statements for a multimap F: X ( Y are equivalent. (a) FTis minimal transfer T closed and (b) m-Cl(F(x)) = F(x) = ;, x2X
T
F(x) = ;,
x2X
x2X
(c) (m–Cl(F))*(y) 5 ; for each y 2 Y, S (d) Y = m-Int(Fc)(x). x2X
Proof. ðaÞ () ðbÞ and ðcÞ ()TðdÞ are straightforward and hence we only prove the implication ðbÞ () ðcÞ. Suppose T m–Cl(F(x)) = F(x) = ; implies that, there is x 2 X in which y R m-Cl(F(x)) and so (b) satisfies. For each y 2 Y, x2X
x2X
x 2 (m-Cl(F))*(y). Therefore, (m–Cl(F))*(y) 5 ; for each y 2 Y. Conversely, suppose (m–Cl(F))*(y) 5 ; for each yT2 Y. Then, for Teach y 2 Y, there is x 2 X for which x 2 (m–Cl(F))*(y) and so y R m–Cl(F(x)) which implies that m–Cl(F(x)) = F(x) = ;. h x2X
x2X
Theorem 2.8. Suppose F, G: X ( Y are two multimaps such that (a) F(x) T G(x) for each x 2 X, G(x) = ;, (b) x2X
(c) G is minimal transfer closed. Then F is minimal transfer closed.
M. Alimohammady et al. / Chaos, Solitons and Fractals 40 (2009) 1162–1168
1167
Proof. Consider x 2 X and y 2 Y with y R F(x). (b) implies there is x0 2 X for which y R G(x0). Since G is minimal transfer closed, so there exists x1 2 X such that y R m–Cl(G(x1)). Now, (a) and part (c) of Proposition 1.1 imply that y R m– Cl(F(x1)); i.e., F is minimal transfer closed. h Lemma 2.2. For a multimap F: X ( Y, F has a maximal element if and only if
T
F(x) ( Y.
x2X
Proof. To prove this, we show that F does not have maximal element if and only if Y =
S
F(x).
x2X
F does not have maximal element () 8y 2 Y F ðyÞ–; () 8y 2 Y 9xy 2 X xy 2 F ðyÞ () 8y 2 Y 9xy 2 X [ () F ðxÞ ¼ Y
y 2 F ðxy Þ
x2X
Corollary 2.2. Suppose F, G: X ( Y are two multimaps satisfying (a) F*(y) G*(y) for each y 2 Y, (b) G* is minimal transfer closed. Then one of the following statements hold. (1) G has maximal element, (2) (m–Int(F)) does not have maximal element. T Proof. Suppose that G has no maximal element, so it follows from part (d) of Lemma 2.1 that G*(y) = ; and hence y2Y T * F (y) = ;. By Theorem 2.8, F* is minimal transfer closed multimap. Then Theorem 2.7 implies that X = y2Y S S m–Int((F*)c)(y). Therefore X = m–Int(F(y)), and so Lemma 2.2 implies that (m–Int(F)) does not have maximal y2Y
y2Y
element, which it completes the proof. h Remark 2.3. We note that Proposition 2.3, Theorems 2.5, 2.6 and 2.7 are generalized versions of similar results in topological spaces (see [5,17–20,27]).
3. Conclusions Minimal structure and minimal space are natural extensions of topology and topological space respectively. Since topology has very important applications in applied sciences, so studying of minimal structure is important from this point of view. The work presented in our paper, not initiate new classes with respect to topology but in view of minimal structures. By considering the spacetime as the product of two minimal structures, the structure of space and that of the spacetime will open the way for new line of research in the field of quantum gravity initiated by Witten and El-Naschie. We expert that this study may extends the accuracy of the approximation in many fields of applications such as e1 theory [9,14], quantum gravity [11,28], discussion making [23], image processing [16], digital topology [15], and others [25,26].
References [1] Alimohammady M, Roohi M, Delavar MR. Knaster-Kuratowski-Mazurkiewicz theorem in minimal generalized convex spaces, Nonlinear Funct Anal Appl, in press. [2] Alimohammady M, Roohi M. Extreme points in minimal spaces. Chaos, Solitons & Fractals 2009;39(3):1480–5. [3] Alimohammady M, Roohi M. Fixed point in minimal spaces. Nonlinear Anal Model Contr 2005;10(4):305–14.
1168
M. Alimohammady et al. / Chaos, Solitons and Fractals 40 (2009) 1162–1168
[4] Alimohammady M, Roohi M. Linear minimal space. Chaos, Solitons & Fractals 2007;33(4):1348–54. [5] Chang SS, Lee BS, Wu X, Cho YJ, Lee M. On the generalized quasi variational inequality problems. J Math Anal Appl 1996;203:686–711. [6] Dinamerico P, Pombo Jr. On a universal property of the final topology. Chaos, Solitons & Fractals 2007;32:212–4. [7] Kozae AbdEl-monem M. On topology expansions by ideals and applications. Chaos, Solitons & Fractals 2002;13:55–60. [8] El Naschie MS. On the uncertainty of Cantorian geometry and two-slit experiment: generalized continuity. Chaos, Solitons & Fractals 1998;9(3):517–29. [9] El Naschie MS. On the unification of heterotic strings, M theory and e(1). Chaos Solitions & Fractals 2000;11(14):2397–408. [10] El Naschie MS. Non-Euclidean spacetime structure and the two slit experiment. Chaos, Solitons & Fractals 2005;26:1–6. [11] El Naschie MS. Superstrings, knots, and noncommutative geometry in space. Int J Theor. Phys 1998;37(12):2935–51. [12] El Naschie MS. Wild topology, hyperbolic geometry and fusion algebra of high energy particle physics. Chaos, Solitons & Fractals 2002;13:1935–45. [13] El Naschie MS. Quantum loops, wild topology and fat Cantor sets in transfinite high-energy physics. Chaos, Solitons & Fractals 2002;13:1167–74. [14] El Naschie MS. Determining the temperature of the microwave background radiation from the topology and geometry of spacetime. Chaos, Solitons & Fractals 2002;14:1121–6. [15] Kong TY, Kopperman R, Meyer PR. A topological approach to digital topology. Am Math Mon 1991;98:901–17. [16] Kong TY, Kopperman R, Meyer PR. Guest editors preface to special issue on digital topology. Topol Appl 1992;46:173–9. [17] Lan KQ. A generalization of H-space and some results on multivalued mappings without convexity. J Math Anal Appl 1995;194:511–28. [18] Lan KQ. New fixed-point theorems for two maps and applications to problems on sets with convex sections and minimax inequalities. Comput Math Appl 2004;47:195–205. [19] Lan KQ, Wu JH. A fixed-point theorem and applications to problems on sets with convex sections and to Nash equilibria. Math Comput Model 2002;36:139–45. [20] Lin LJ. Applications of a fixed point theorem in G-convex space. Nonlinear Anal 2001;46:601–8. [21] Maki H. On generalizing semi-open sets and preopen sets, In: Proceedings of the meeting on topological spaces theory and its application; August 1996. p. 13–8. [22] Maki H, Umehara J, Noiri T. Every topological space is pre T 12 . Mem Fac Sci Kochi Univ Ser A Math 1996;17:33–42. [23] Pawlak Z. Raugh sets, Raugh relations and Raugh functions. Bull Pol Ac Math 1996:15–9. [24] Popa V, Noiri T. On M-continuous functions, Anal Univ Dunarea Jos-Galati. Ser Mat Fiz Mec Teor Fasc II 2000;18(23):31–41. [25] Shuliang W, Deren LI, Wenzhong SHI, Xinzhou W, Deyi LI. Raugh spatial description. I.A.P.R.S. 2002;44(2):20–3. [26] Stadler BMR, Stadler PF. Generalized topological spaces in evolutionary theory and combinatorial chemistry. J Chem Inform Comput Sci 2002;42:577–85. [27] Tian CQ. Generalization of FKKM theorem and the Ky Fan minimax inequality with applications to maximal elements, price equilibrium and complementarity. J Math Anal Appl 1992;170:457–71. [28] Witten E. Reflections on the fate of spacetime. Phys Today 1996:24–30.
Transfer closed and transfer open multimaps in minimal spaces M. Alimohammady *, M. Roohi, M.R. Delavar Department of Mathematics, University of Mazandaran, Babolsar 47416-1468, Iran Accepted 29 August 2007
Abstract This paper is devoted to introduce the concepts of transfer closed and transfer open multimaps in minimal spaces. Also, some characterizations of them are considered. Further, the notion of minimal local intersection property will be introduced and characterized. Moreover, some maximal element theorems via minimal transfer closed multimaps and minimal local intersection property are given. 2007 Elsevier Ltd. All rights reserved.
1. Introduction and preliminaries It is well known that topological concepts have many applications in modern physics. For example, the topology of quantum spacetime is shadowed closely by the Mobius geometry of quasi-Fuschian and Kleinian groups and that is the cause behind the phenomena of high-energy particle physics [11]. In fact, considering the spacetime as the product of two topologies, the topology of space and that of the spacetime will open the way for new line of research in the field of quantum gravity initiated by Witten and El-Naschie. For the relation between Wild Topology, Hyperbolic Geometry and Fusion Algebra with Coupling constants of the standard model and quantum gravity and close connection between e1 theory and the topological theory of four manifolds, we refer to [12,13] and for the relation between topological concepts and geometrical properties of the e1 spacetime to [14]. For another example, constructing a topology via a relation on a real-life data will help in mathematizing many fields. In fact, if X is a collection of symptoms and diseases in a certain region and R is a binary relation on X given by an expert the topology on X generated by R is a knowledge base for X, indication of symptoms for a fixed disease can be seen through the topology [7]. Since topology has very important applications in applied sciences, so studying of minimal structure as a generalization of topology is important from this point of view. Minimal structures may have very important applications in quantum particles physics, particularly in connection with string theory and e1 theory [6,8–10]. The work presented
*
Corresponding author. E-mail addresses: [email protected] (M. Alimohammady), [email protected] (M. Roohi), [email protected] (M.R. Delavar). 0960-0779/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.08.071
M. Alimohammady et al. / Chaos, Solitons and Fractals 40 (2009) 1162–1168
1163
in our paper, not initiate new classes with respect to topology but in view of minimal structures. In this paper after introducing transfer closed and transfer open multimaps via minimal space, some characterizations of them are given. Moreover, the concept of minimal local intersection property will be introduced and some its characterization will be investigated. Finally, some maximal element theorems via minimal transfer closed multimaps and minimal local intersection property are considered. The concepts of minimal structure and minimal spaces, as a generalization of topology and topological spaces were introduced in [21]. For easy understanding of the material incorporated in this paper we recall some basic definitions. For details on the following notions we refer to [1–4,21,22] and [24] and references therein. A family M # PðX Þ is said to be a minimal structure on X if ;, X 2 M. In this case ðX ; MÞ is called a minimal space. For some examples in this setting see [21,24]. In a minimal space ðX ; MÞ, A 2 PðX Þ is said to be an m-open set if A 2 M and also B 2 PðX Þ is an m-closed set if Bc 2 M. For any x 2 X, N(x) is said to be a minimal neighborhood of x, if for any S z 2 N(x) there is an m-open subset G N(x) such that z 2 G . Set m IntðAÞ ¼ fU : U # A; U 2 Mg and z z T m ClðAÞ ¼ fF : A # F ; F c 2 Mg. Proposition 1.1. [21] For any two sets A and B, (a) (b) (c) (d) (e) (f) (g)
m-Int(A) A and m-Int(A) = A if A is an m-open set, A m-Cl(A) and A = m-Cl(A) if A is an m-closed set, m-Int(A) m-Int(B) and m- Cl(A) m-Cl(B) if A B, m-Int(A \ B) = (m-Int(A)) \ (m-Int(B)) and (m-Int(A)) [ (m-Int(B)) m-Int(A [ B), m-Cl(A [ B) = (m-Cl(A)) [ (m-Cl(B)) and m -Cl(A \ B) (m-Cl(A)) \ (m-Cl(B)), m-Int (m-Int(A)) = m-Int(A) and m–Cl(m–Cl(B)) = m-Cl(B), (m-Cl(A))c = m-Int(Ac) and (m-Int(A))c = m-Cl(Ac).
2. Main Results Suppose X and Y are two minimal spaces. A multimap F: X ( Y is a function from a set X into the power set of Y; that is, a function with the values F(x) Y for all x 2 X. Given A X, set [ F ðxÞ: F ðAÞ ¼ x2A
We say that a multimap F: X ( Y has a maximal element if F(x0) = ;, for some x0 2 X. For a multimap F: X ( Y, the multimaps Fc, m–Cl(F), and m–Int(F) from X to Y are defined by Fc(x) = {y 2 Y: y R F(x)}, (m–Cl(F))(x) = m–Cl(F(x)) and (m–Int(F))(x) = m–Int(F(x)), respectively. Also multimaps F and F* from Y to X are defined by F(y) = {x 2 X:y 2 F(x)} and F*(y) = {x 2 X:y R F(x)}, respectively. Some properties of these multimaps and their relations can be found in [17,19]. Lemma 2.1. [19] Suppose F, G: X ( Y are two multimaps. Then (a) (b) (c) (d) (e) (f)
for each x 2 X, F(x) G(x) if and only if G*(y) F*(y) for each y 2 Y, y R F(x) if and only if x 2 F*(y), for each x 2 X, (F*)*(x) = F(x), T for each x 2 X, F(x) 5 ; if and only if F*(y) = ;, y2A for each y 2 Y, (Fc)*(y) = F(y), c * for each y 2 Y, (F ) (y) = F (y).
Proposition 2.1. Suppose F: X ( Y is a multimap. Then (a) (b) (c) (d)
(m–Cl(F))c = m–Int(Fc), (m–Cl(F))* = (m– Int(Fc)), (m–Cl(F*))c = m–Int(F), (m–Cl(F*))* = (m–Int(F)).
Proof. It is an immediate consequence of Proposition 1.1 and Lemma 2.1.
h
1164
M. Alimohammady et al. / Chaos, Solitons and Fractals 40 (2009) 1162–1168
Definition 2.1. Suppose X and Y are two minimal spaces. A multimap F: X ( Y is said to be (a) minimal transfer open if for each x 2 X and y 2 F(x), there exists x0 2 X for which y 2 m–Int(F(x0)), (b) minimal transfer closed if for any x 2 X and y R F(x), there exists x0 2 X for which y R m–Cl(F(x0)). Theorem 2.1. Suppose X and Y are two minimal spaces. The following are equivalent. (a) The S F: X ( Y is minimal transfer open, S multimap F(x) = m–Int(F(x)), (b) x2X x2A T T c F (x) = m–Cl(Fc(x)), (c) x2X x2X T T c F (x) = (m-Int(F(x)))c. (d) x2X
x2X
Proof. The implications ðbÞ $ ðcÞ, ðcÞ $ ðdÞ and ðbÞ ! ðaÞ are straightforward. OnlySwe must prove the S implication ðaÞ ! ðbÞ. For this, assume (a) is true. It follows from part (a) of Proposition 1.1 that m–Int(F(x)) F(x). On the x2X x2X S other hand, for y 2 F(x), there is x1 2 X such that y 2 F(x1). Then from (a) there is x0 2 X for which y 2 m–Int(F(x0)) S Sx2X and so F(x) m–Int(F(x)); i.e., (b) satisfies, which it completes the proof. h x2X
x2X
Theorem 2.2. Suppose X and Y are two minimal spaces. A multimap F: X ( Y is minimal transfer open if and only if Fc is minimal transfer closed. Proof. Suppose F is minimal transfer open. Consider x 2 X and y 2 Y in which y R Fc(x); i.e., y 2 F(x). By the assumption, there is x0 2 X for which y 2 m–Int(F(x0)). Therefore, y R m–Cl(Fc(x0)) and so Fc is minimal transfer closed. Conversely, suppose Fc is minimal transfer closed. For any x 2 X, if y 2 F(x) then y R Fc(x). From the assumption there is x0 2 X such that y R m–Cl(Fc(x0)). Therefore, y 2 m–Int(F(x0)). h Theorem 2.3. Suppose X and Y are two minimal spaces. The following are equivalent. (a) The T F:X ( Y is minimal transfer closed, T multimap F(x) = m–Cl(F(x)), (b) x2X x2X S S c F (x) = m–Int(Fc(x)), (c) x2X x2X S S c F (x) = (m–Cl(F(x)))c. (d) x2X
x2X
Proof. It is an immediate consequence of Theorems 2.1 and 2.2.
h
Remark 2.1. Theorems 2.2 and 2.3 are extended versions of Lemma 2.1 in [5]. Definition 2.2. Suppose X and Y are two minimal spaces. A multimap F:X ( Y is said to have minimal local intersection property if for each x 2 X with F(x) 5 ;, there is minimal neighborhood N(x) of x for which \ F ðyÞ–;: y2mIntðN ðxÞÞ
Proposition 2.2. Suppose X and Y are two minimal spaces. Then F: X ( Y has the minimal local intersection property if and only if for each x 2 F(y), there is a minimal neighborhood N(x) of x and y1 2 Y for which m– Int(N(x)) F(y1). Proof. Suppose F has the minimal local intersection property and that x 2 F(y). Therefore, y 2 F(x); i.e. F(x) 5 ; and so by the assumption there is a minimal neighborhood N(x) of x for which \ F ðyÞ–;: y2mIntðN ðxÞÞ
M. Alimohammady et al. / Chaos, Solitons and Fractals 40 (2009) 1162–1168
1165
Then there is y1 2 Y with y1 2 F(z) for all z 2 m–Int(N(x)). Consequently, m-Int(N(x)) F(y1). Conversely, suppose F(x) 5 ;, there is y 2 Y for which x 2 F(y). There exits a minimal neighborhood N(x) of x and y1 2 Y in which m–Int(N(x)) F(y1). Therefore, z 2 F(y1) for all z 2 m–Int(N(x)) which completes the proof. h Theorem 2.4. Suppose X and Y are two minimal spaces. Then F: X ( Y has the minimal local intersection property if and only if F: Y ( X is minimal transfer open. Proof. Suppose F has the minimal local intersection property and x 2
S
F(y). There is y0 2 Y in which x 2 F(y0) and
y2Y
so y0 2 F(x). According to Proposition 2.2 there is minimal neighborhood N(x) of x and y1 2 Y in which mInt(N(x)) F(y1). Therefore, x 2 m IntðN ðxÞÞ # m IntðF ðy 1 ÞÞ #
[
m IntðF ðyÞÞ;
y2Y
i.e., F is minimal transfer open. Conversely, suppose that F is minimal transfer open. Consider an arbitrary element x 2 X with F(x) 5 ;. There is y1 2 F(x) and so x 2 F(y1). Since F is minimal transfer open, so Theorem 2.1 implies that, there isT an element y0 2 Y such that x 2 m–Int(F(y0)). Now, if z 2 m–Int(F(y0)) then z 2 F(y0) and so y0 2 F(z). Hence, F ðzÞ–;. h z2mIntðF ðy 0 ÞÞ
Corollary 2.1. Suppose X and Y are two minimal spaces. Then (a) F: X ( Y has the minimal local intersection property if and only if F*:Y ( X is minimal transfer closed, (b) F: X ( Y is minimal transfer closed if and only if F*:Y ( X has the minimal local intersection property. Proof. The first assertion follows from Theorem 2.4 and part (f) of Lemma 2.1. Also, the second part immediately follows from (c) in Lemma 2.1 and (a) in this corollary. h Remark 2.2. It should be noticed that the origin versions of Proposition 2.2, Theorem 2.4 and Corollary 2.1 go back to Lan and Wu [19]. Proposition 2.3. Suppose X and Y are two minimal spaces. Then the following statements for a multimap F: X ( Y are equivalent. (a) F: Y ( X is minimal transfer open and F(x) 5 ; for all x 2 X, (b) F: X ( Y has the minimal local intersection property and F(x) 5 ; for all x 2 X, (c) X = ¨{m–Int(F(y)): y 2 Y}. Proof. That (a) and (b) are equivalent follows from Theorem 2.4. Suppose (b) satisfies and x 2 X is arbitrary. Since F(x) 5 ; so there is a minimal neighborhood N(x) of x for which \ F ðyÞ–;: y2mIntðN ðxÞÞ
There exists y0 2 Y such that y0 2 F(z) for all z 2 m–Int(N(x)); i.e., z 2 F(y0) for all z 2 m–Int(N(x)). Therefore, x 2 m IntðN ðxÞÞ # m IntðF ðy 0 ÞÞ; and so x 2 m–Int(F(y0)). S S For implication ðcÞ ) ðaÞ, suppose X = ¨{m–Int(F(y)): y 2 Y}. We must show that F (y) = m–Int(F(y)). y2Y y2Y One direction is straightforward, and other follows from the fact that [ [ F ðyÞ # X ¼ m IntðF ðyÞÞ: y2Y
y2Y
Also for each x 2 X, there is y0 2 Y for which x 2 m-Int(F(y0)) and so x 2 F(y0). Then y0 2 F(x); i.e., F(x) 5 ; for all x 2 X. h
1166
M. Alimohammady et al. / Chaos, Solitons and Fractals 40 (2009) 1162–1168
Theorem 2.5. Suppose X and Y are two minimal spaces. For a multimap F: X ( Y the following statements are equivalent. (a) FThas the minimalTlocal intersection property and F(x) 5 ; for all x 2 X, m-Cl(F*(y)) = F*(y) = ;, (b) y2Y
y2Y
(c) (m-Cl(F*))*(x) 5 ; for each x 2 X, S (d) X = m–Int(F(y)). y2Y
Proof. ðaÞ ) ðbÞ follows from Corollary 2.1, (a) and (b) in Theorem 2.3 and (d) in Lemma 2.1. Suppose T m–Cl(F*(y)) = ;, so for any x 2 X there exists y 2 Y such that x R m–Cl(F*(y)). This mean that y 2 (m–Cl(F*))*(x). y2Y
Therefore for any x 2 X, (m–Cl(F*))*(x) 5 ; and so ðbÞ ) ðcÞ is proved. For ðcÞ ) ðdÞ, suppose that (c) is true. So *))*(x). Part (d) in Proposition 2.1, implies that y 2 (m–Int(F))(x) for any x 2 X there is y 2 Y in which y 2 (m–Cl(F S and therefore x 2 (m–Int(F )(y)). Hence X = m–Int(F)(y) and so the implication ðcÞ ) ðdÞ is proved. Finally, supy2Y T T S pose that X = m–Int(F)(y), so (m-Int(F))c(y) = ;. Condition (c) of Proposition 2.1 shows that F*(y) = ; and y2Y
y2Y
y2Y
from (d) in Lemma 2.1, F(x) 5 ; for any x 2 X. Also from Theorem 2.3 and Corollary 2.1, F has the minimal local intersection property. h Theorem 2.6. Suppose F, G: X ( Y are two multimaps satisfying (a) for each x 2 X, F(x) G(x), (b) F has the minimal local intersection property. Then one of the following statements hold. (1) G has the minimal local intersection property, (2) F has a maximal element. Proof. Suppose x 2 X with G(x) 5 ;. If F has no maximal element, then condition (b) implies that there is a minimal neighborhood N(x) of x for which ˙{F(z): z 2 m–Int(N(x))} 5 ; and hence by (a) we have ˙{G(z): z 2 m– Int(N(x))} 5 ;; i.e., (1) holds. h Theorem 2.7. The following statements for a multimap F: X ( Y are equivalent. (a) FTis minimal transfer T closed and (b) m-Cl(F(x)) = F(x) = ;, x2X
T
F(x) = ;,
x2X
x2X
(c) (m–Cl(F))*(y) 5 ; for each y 2 Y, S (d) Y = m-Int(Fc)(x). x2X
Proof. ðaÞ () ðbÞ and ðcÞ ()TðdÞ are straightforward and hence we only prove the implication ðbÞ () ðcÞ. Suppose T m–Cl(F(x)) = F(x) = ; implies that, there is x 2 X in which y R m-Cl(F(x)) and so (b) satisfies. For each y 2 Y, x2X
x2X
x 2 (m-Cl(F))*(y). Therefore, (m–Cl(F))*(y) 5 ; for each y 2 Y. Conversely, suppose (m–Cl(F))*(y) 5 ; for each yT2 Y. Then, for Teach y 2 Y, there is x 2 X for which x 2 (m–Cl(F))*(y) and so y R m–Cl(F(x)) which implies that m–Cl(F(x)) = F(x) = ;. h x2X
x2X
Theorem 2.8. Suppose F, G: X ( Y are two multimaps such that (a) F(x) T G(x) for each x 2 X, G(x) = ;, (b) x2X
(c) G is minimal transfer closed. Then F is minimal transfer closed.
M. Alimohammady et al. / Chaos, Solitons and Fractals 40 (2009) 1162–1168
1167
Proof. Consider x 2 X and y 2 Y with y R F(x). (b) implies there is x0 2 X for which y R G(x0). Since G is minimal transfer closed, so there exists x1 2 X such that y R m–Cl(G(x1)). Now, (a) and part (c) of Proposition 1.1 imply that y R m– Cl(F(x1)); i.e., F is minimal transfer closed. h Lemma 2.2. For a multimap F: X ( Y, F has a maximal element if and only if
T
F(x) ( Y.
x2X
Proof. To prove this, we show that F does not have maximal element if and only if Y =
S
F(x).
x2X
F does not have maximal element () 8y 2 Y F ðyÞ–; () 8y 2 Y 9xy 2 X xy 2 F ðyÞ () 8y 2 Y 9xy 2 X [ () F ðxÞ ¼ Y
y 2 F ðxy Þ
x2X
Corollary 2.2. Suppose F, G: X ( Y are two multimaps satisfying (a) F*(y) G*(y) for each y 2 Y, (b) G* is minimal transfer closed. Then one of the following statements hold. (1) G has maximal element, (2) (m–Int(F)) does not have maximal element. T Proof. Suppose that G has no maximal element, so it follows from part (d) of Lemma 2.1 that G*(y) = ; and hence y2Y T * F (y) = ;. By Theorem 2.8, F* is minimal transfer closed multimap. Then Theorem 2.7 implies that X = y2Y S S m–Int((F*)c)(y). Therefore X = m–Int(F(y)), and so Lemma 2.2 implies that (m–Int(F)) does not have maximal y2Y
y2Y
element, which it completes the proof. h Remark 2.3. We note that Proposition 2.3, Theorems 2.5, 2.6 and 2.7 are generalized versions of similar results in topological spaces (see [5,17–20,27]).
3. Conclusions Minimal structure and minimal space are natural extensions of topology and topological space respectively. Since topology has very important applications in applied sciences, so studying of minimal structure is important from this point of view. The work presented in our paper, not initiate new classes with respect to topology but in view of minimal structures. By considering the spacetime as the product of two minimal structures, the structure of space and that of the spacetime will open the way for new line of research in the field of quantum gravity initiated by Witten and El-Naschie. We expert that this study may extends the accuracy of the approximation in many fields of applications such as e1 theory [9,14], quantum gravity [11,28], discussion making [23], image processing [16], digital topology [15], and others [25,26].
References [1] Alimohammady M, Roohi M, Delavar MR. Knaster-Kuratowski-Mazurkiewicz theorem in minimal generalized convex spaces, Nonlinear Funct Anal Appl, in press. [2] Alimohammady M, Roohi M. Extreme points in minimal spaces. Chaos, Solitons & Fractals 2009;39(3):1480–5. [3] Alimohammady M, Roohi M. Fixed point in minimal spaces. Nonlinear Anal Model Contr 2005;10(4):305–14.
1168
M. Alimohammady et al. / Chaos, Solitons and Fractals 40 (2009) 1162–1168
[4] Alimohammady M, Roohi M. Linear minimal space. Chaos, Solitons & Fractals 2007;33(4):1348–54. [5] Chang SS, Lee BS, Wu X, Cho YJ, Lee M. On the generalized quasi variational inequality problems. J Math Anal Appl 1996;203:686–711. [6] Dinamerico P, Pombo Jr. On a universal property of the final topology. Chaos, Solitons & Fractals 2007;32:212–4. [7] Kozae AbdEl-monem M. On topology expansions by ideals and applications. Chaos, Solitons & Fractals 2002;13:55–60. [8] El Naschie MS. On the uncertainty of Cantorian geometry and two-slit experiment: generalized continuity. Chaos, Solitons & Fractals 1998;9(3):517–29. [9] El Naschie MS. On the unification of heterotic strings, M theory and e(1). Chaos Solitions & Fractals 2000;11(14):2397–408. [10] El Naschie MS. Non-Euclidean spacetime structure and the two slit experiment. Chaos, Solitons & Fractals 2005;26:1–6. [11] El Naschie MS. Superstrings, knots, and noncommutative geometry in space. Int J Theor. Phys 1998;37(12):2935–51. [12] El Naschie MS. Wild topology, hyperbolic geometry and fusion algebra of high energy particle physics. Chaos, Solitons & Fractals 2002;13:1935–45. [13] El Naschie MS. Quantum loops, wild topology and fat Cantor sets in transfinite high-energy physics. Chaos, Solitons & Fractals 2002;13:1167–74. [14] El Naschie MS. Determining the temperature of the microwave background radiation from the topology and geometry of spacetime. Chaos, Solitons & Fractals 2002;14:1121–6. [15] Kong TY, Kopperman R, Meyer PR. A topological approach to digital topology. Am Math Mon 1991;98:901–17. [16] Kong TY, Kopperman R, Meyer PR. Guest editors preface to special issue on digital topology. Topol Appl 1992;46:173–9. [17] Lan KQ. A generalization of H-space and some results on multivalued mappings without convexity. J Math Anal Appl 1995;194:511–28. [18] Lan KQ. New fixed-point theorems for two maps and applications to problems on sets with convex sections and minimax inequalities. Comput Math Appl 2004;47:195–205. [19] Lan KQ, Wu JH. A fixed-point theorem and applications to problems on sets with convex sections and to Nash equilibria. Math Comput Model 2002;36:139–45. [20] Lin LJ. Applications of a fixed point theorem in G-convex space. Nonlinear Anal 2001;46:601–8. [21] Maki H. On generalizing semi-open sets and preopen sets, In: Proceedings of the meeting on topological spaces theory and its application; August 1996. p. 13–8. [22] Maki H, Umehara J, Noiri T. Every topological space is pre T 12 . Mem Fac Sci Kochi Univ Ser A Math 1996;17:33–42. [23] Pawlak Z. Raugh sets, Raugh relations and Raugh functions. Bull Pol Ac Math 1996:15–9. [24] Popa V, Noiri T. On M-continuous functions, Anal Univ Dunarea Jos-Galati. Ser Mat Fiz Mec Teor Fasc II 2000;18(23):31–41. [25] Shuliang W, Deren LI, Wenzhong SHI, Xinzhou W, Deyi LI. Raugh spatial description. I.A.P.R.S. 2002;44(2):20–3. [26] Stadler BMR, Stadler PF. Generalized topological spaces in evolutionary theory and combinatorial chemistry. J Chem Inform Comput Sci 2002;42:577–85. [27] Tian CQ. Generalization of FKKM theorem and the Ky Fan minimax inequality with applications to maximal elements, price equilibrium and complementarity. J Math Anal Appl 1992;170:457–71. [28] Witten E. Reflections on the fate of spacetime. Phys Today 1996:24–30.