- Email: [email protected]
Linear minimal space
Chaos, Solitons and Fractals 33 (2007) 1348–1354 www.elsevier.com/locate/chaos
Linear minimal space M. Alimohammady a
a,*
, M. Roohi
b
Department of Mathematics, University of Mazandaran, Babolsar, Iran b Islamic Azad University, Sari-branch, Iran Accepted 23 January 2006
Abstract This paper deals with minimal linear spaces, m-continuity and m-boundedness. In particular, it is found that in a linear minimal space (X ; M) the assignment x # t0x + x0 from X to X is m-continuous. On the other hand, the convex hull of an m-neighborhood of 0 is an m-neighborhood if (X ; M) has property U. 2006 Elsevier Ltd. All rights reserved.
1. Introduction and preliminaries This paper deals with minimal structure as an extended notion of topology via linear spaces which has many applications in applied sciences such as quantum gravity [5,21], discussion making [12], image processing [8], digital topology [7], and others [14,15]. The topology of quantum spacetime is shadowed closely by the Mobius geometry of quasiFuschian and Kleinian groups and that is the cause behind the phenomena of high-energy particle physics [5]. 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. Minimal structures may have very important applications in quantum particles physics, particularly in connection with string theory and e1 theory [2–4]. One of conclusions of advanced scientific research into the very basic question related to the quintessence of natural science and philosophy is topology and can be minimal structure. For example, topology has many applications in string theory, in the study of DNA replication and recombination, and in areas of statistical mechanics. DNA topology is the focus of a subdiscipline within molecular biology and as a term refers to both the knot-like arrangements that segments of DNA may assume and to the mathematics that pertains to them. The topology of DNA topoisomers is important to replication, transcription and recombination, including the recombination events important to the life cycles of many viruses. Topoisomerases are enzymes that change the topology of DNA; for more details we refer to [16,18–20]. Many properties on topological spaces such as boundedness and continuity are discussed via minimal notion. While others are far from being completely devoted in its foundation. So, this paper is devoted to study the class of minimal structures via linear spaces. Some characterizations and several properties of these concepts are discussed.
*
Corresponding author. E-mail addresses: [email protected] (M. Alimohammady), [email protected] (M. Roohi).
0960-0779/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.01.100
M. Alimohammady, M. Roohi / Chaos, Solitons and Fractals 33 (2007) 1348–1354
1349
In [10] authors introduced minimal structures and minimal spaces. Some results about minimal spaces can be found in [1,6,9,11,13]. As is well known, a topological vector space is a vector space endowed with a topology for which two operations + : X · X ! X ((x,y) # x + y) and : F X ! X ððt; xÞ 7! txÞ are continuous. In what follows we will consider linear minimal vector space, it is shown that x # t0x + x0 is m-continuous, further the convex hull of an m-neighborhood of 0 is an m-neighborhood if (X ; M) has the property U. For easy understanding of the material incorporated in this paper we recall some basic definitions. For details on the following notions we refer to [9,13,17]. Definition 1.1 [10]. A family M PðX Þ is said to be a minimal structure on X if ;; X 2 M. In this case (X ; M) is called a minimal space. Example 1.2 [10]. Let (X, s) be a topological space. Then, M ¼ s; SOðX Þ; POðX Þ; aOðX Þ and bO(X) are minimal structures on X. Definition 1.3 [10]. A set A 2 PðX if A 2 M B 2 PðX Þ is an m-closed set S Þ is said to be an m-open set (m-neighborhood) T if Bc 2 M. We set m IntðAÞ ¼ fU : U A; U 2 Mg and m ClðAÞ ¼ fF : A F ; F c 2 Mg. Proposition 1.4 [10]. 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).
Definition 1.5 [13]. A minimal space (X ; M) enjoys the property U if ‘‘the arbitrary union of m-open sets is m-open’’. (X ; M) has the property I if ‘‘the any finite intersection of m-open sets is m-open’’. Proposition 1.6 [13]. For a minimal structure M on a set X, the following are equivalent: (a) M has the property U. (b) If m Int(A) = A, then A 2 M. (c) If m Cl(B) = B, then Bc 2 M.
2. Minimal product spaces In the following section we introduce minimal structure for product of minimal spaces. Lemma 2.1. The intersection and union of any collection of minimal structures is a minimal structure. Proof. It is straightforward.
h
Definition 2.2. f : ðX ; MÞ ! ðY ; NÞ is said to be minimal continuous (briefly m-continuous) if f1(U) is an m-open set, where U is any m-open set in N. Theorem 2.3. The composition of two m-continuous functions is an m-continuous function. Proof. It is direct.
h
In the next result we would like to introduce product minimal structure. First we will consider the initial minimal structure for a family of functions.
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Proposition 2.4. Suppose ðY ; NÞ is a minimal space and f : X ! ðY ; NÞ is a function. Then there is a weakest minimal structure on X for which f is m-continuous. Proof. The required minimal structure is M ¼ f 1 ðNÞ ¼ ff 1 ðV Þ : V 2 Ng.
h
Corollary 2.5. Suppose ðX ; MÞ is a minimal structure and Y X. Then there is a weakest minimal structure on Y say N, such that the inclusion map i : ðY ; NÞ ! ðX ; MÞ is m-continuous. Proof. It is enough to consider i instead of f in Proposition 2.4
h
We call N the induced minimal structure by M on Y. Corollary 2.6. Suppose Y X and f : ðX ; MÞ ! ðZ; PÞ is m-continuous. Then f jY is m-continuous if Y is endowed with induced minimal structure. Proof. That f jY is m-continuous follows from f jY = foi and Theorem 2.3, where i : Y ! X is the inclusion map.
h
Theorem 2.7. Consider a family fðX a ; Na Þ : a 2 Ag of minimal spaces and ffa : X ! ðX a ; Ma Þ : a 2 Ag a family of functions. Then there is a weakest minimal structure M on X such that fa’s are m-continuous. Proof. It is easy to see that M ¼ Na ; a 2 Ag. h
S
a Ma
is the required minimal structure, where Ma ¼ f 1 ðNa Þ ¼ ffa1 ðV Þ : V 2
Theorem 2.8. Consider a family ffa : X ! ðX a ; Ma Þ : a 2 Ag of functions, where ðX a ; Ma Þ are minimal spaces. Equipped X by minimal structure M generated by ffa : a 2 Ag. f : ðY ; NÞ ! ðX ; MÞ is m-continuous if and only if fa f is m-continuous function for all a 2 A. Proof. If f is m-continuous function, then from Theorem 2.3 for each a 2 A; fa f is m-continuous. For the converse, suppose on the contrary fa f’s are m-continuous functions for each a 2 A but f is not m-continuous. Hence, there is B 2 M such that f 1 ðBÞ 62 N. Then one of the following two statements holds: (i) 9a0 2 A; 9Ba0 2 Ma0 such that B ¼ fa01 ðB0 Þ. (ii) 8a 2 A; 8Ba 2 Ma ; B 6¼ f 1 ðBa Þ. 1 ðB0 ÞÞ ¼ ðfa0 f Þ1 ðBa0 Þ, Therefore, ðfa0 f Þ1 ðBa0 Þ 62 N which is a In case (i), we have f 1 ðBÞ ¼ f 1 ðfa0 contradiction, because f is an m-continuous function. In the other case, since f1(;) = ; and f1(Y) = X so, B 62 {;, Y}. Then M n fBg is a minimal structure on X and so for each a 2 A; fa : ðX ; M n fBgÞ ! ðY ; NÞ is m-continuous, which is a contradiction according to the choice of M. h
As a consequence of Theorem 2.7 we present the product minimal structure for an arbitrary family fðX a ; Ma Þ : a 2 Ag of minimal spaces. Product minimal structure on X ¼ Pa2A X a is the weakest minimal structure on X (denoted by M ¼ Pa Ma Þ, such that for each b 2 A the canonical projection pb : Pa2A X a ! X b is m-continuous. Corollary 2.9. For any family fðX a ; Ma Þ : a 2 Ag of minimal spaces, product minimal structure on X ¼ Pa2A X a exists. Proof. Only we must consider pa in Theorem 2.7 instead of fa.
h
Corollary 2.10. Suppose X ¼ Pa2A X a ; where each Xa is a minimal space. Endow X to the product minimal structure generated by fPa X a : a 2 Ag. Then f is m-continuous function if and only if pa f of is m-continuous for all a 2 A, where f : ðY ; NÞ ! ðX ; MÞ is a function. Proof. It follows immediately from Theorem 2.8
h
Corollary 2.11. Suppose f : ðX ; MÞ ! ðY ; NÞ and g : ðX ; MÞ ! ðZ; PÞ are m-continuous functions. Then the function f g : ðX ; MÞ ! ðY Z; N PÞ defined by (f · g)(x) = (f(x), g(x)) is m-continuous.
M. Alimohammady, M. Roohi / Chaos, Solitons and Fractals 33 (2007) 1348–1354
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Proof. p1(f · g)(x) = p1(f(x), g(x)) = f(x) and p2(f · g)(x) = p2(f(x), g(x)) = g(x). Hence, p1(f · g) and p2(f · g)(x) are mcontinuous. Then Corollary 2.10 implies that f · g is m-continuous. h
3. Linear minimal spaces and m-bounded sets In this section we show that x # t0x + x0 is m-continuous. Definition 3.1. A linear minimal structure on a vector space X over the complex field F is a minimal structure M on X such that the two mappings þ :X X ! X ; ðx; yÞ 7! x þ y :F X ! X ; ðt; xÞ 7! tx are m-continuous, when F has the usual topology and F X and X · X the corresponding product minimal structures. A linear minimal space (or m-linear space) is a vector space together with a linear minimal structure. Proposition 3.2. Suppose f, g : X ! Y are m-continuous functions, where X and Y are minimal space and linear minimal space, respectively. Then f + g : X ! Y is m-continuous function too. Proof. Consider h : Y · Y ! Y which is defined by h(y1, y2) = y1 + y2. That f + g is m-continuous follows from Theorem 2.8, f + g = ho(f · g) and Corollary 2.11. h Theorem 3.3. Let X be a linear minimal space, t0 2 F and x0 2 X. Then f : X ! X defined by f(x) = t0x + x0 is m-continuous. If t0 5 0, then f is a minimal homeomorphism. Proof. Consider f1 : X ! F and f2 : X ! X defined by f1(x) = t0 and f2(x) = x, respectively. It is easy to see that f1 and f2 are m-continuous. Consequently, Corollary 2.10 implies that f1 f2 : X ! F X is m-continuous. Since g : F X ! X defined by g(t, x) = tx is m-continuous, so g (f1 · f2), and f1 · f2 are m-continuous. Then from Proposition 3.2, f = g (f1 · f2) + x0 is m-continuous. For the second assertion only we must note that f 1 ðxÞ ¼ t10 ðx x0 Þ. h Corollary 3.4. If A is an m-open set in a linear minimal space X, and t 5 0, then tA is m-open. Proof. Both tA = f1(A) and m-continuity of f : X ! X defined by f ðxÞ ¼ 1t x complete the proof.
h
Corollary 3.5. Let (X ; M) be a minimal structure, A be an m-open set in X and x0 2 X. Then x0 + A is an m-open set. Proof. Consider g = h (f · idX), where f : ðX ; MÞ ! ðX ; MÞ is defined by f(x) = x0 and h : X · X ! X is defined by h(x, y) = x + y. Similar to the proof of Theorem 3.3, f · idX is m-continuous. Hence, g is m-continuous by Theorem 2.8. That x0 + A is m-open follows from this and the fact that g1(A) = x0 + A. h In the following we deal with m-bounded sets in a linear minimal space. Definition 3.6. A X is called balanced if D Æ A = A. We call D Æ A the balanced hall of A and is denoted by bal(A). Proposition 3.7. Suppose (X ; M) is a fuzzy minimal space with the property U. Then the balanced hall of any m-neighborhood of 0 is also an m-neighborhood of 0. Proof. Since balðU Þ ¼ D U ¼
[
tU
jtj61
and from Theorem 3.3 each t Æ U is m-open, so property U implies that bal(U) is m-open.
h
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Proposition 3.8. m Cl(A) is balanced if A is balanced. Proof. Consider f : F X ! X by f(a, x) = a Æ x is m-continuous, so f(m Cl(D · A)) m Cl(f(D · A)) and so f(D · m Cl(A)) m Cl(f(D · A)). Then D Æ (m Cl(A)) m Cl(D Æ A) = m Cl(A) which implies that m Cl(A) is balanced. h Proposition 3.9. Suppose U is an m-(open)neighborhood of 0 in X and V is the family of m-balanced halls of members of U. Then V is an m-open neighborhood base at 0. Proof. Assume U 2 U. Since 0 + 0 = 0 so there are a P 0 and m-open neighborhood W1 of 0 such that (aD) Æ W1 U. Therefore, W2 = D Æ (aW1) U, so W2 is an m-open balanced neighborhood contained in U. h Definition 3.10. In a minimal vector space ðX ; MÞ; A X is said to be (a) m-bounded set if there is t0 > 0 such that A tU, where U is any m-open set containing 0 and jtj 6 t0, (b) convex if [ t1 V þ þ tn V coðV Þ ¼ Pn 06ti 61;
i¼1
ti ¼1;n2N
is a subset of A.
Proposition 3.11. If A B and B is m-bounded, then A is m-bounded too. Proof. It is straightforward.
h
Proposition 3.12. Suppose A and B are m-bounded, then A [ B is m-bounded too. Proof. Consider U as an m-neighborhood of 0. Prom the assumption there are t1, t2 > 0 such that t 6 min{t1, t2} implies that tA U and tB U. Then t(A [ B) = tA [ tB U. h Proposition 3.13. Suppose (X ; M) has property I, A and B are m-bounded. Then A + B is m-bounded. Proof. Suppose W is any m-open set containing 0, from m-continuity of + there are two m-neighborhoods U1 and U2 of 0 for which U1 + U2 W. Set U1 \ U2 = U. Then U + U W. There are t1,t2 > 0 such that jtj 6 min{t1,t2} implies that tA U and tB U. Then t(A + B) = tA + tB U + U W. h The following result shows the stability of m-bounded sets under m-continuous functions. Proposition 3.14. Suppose ðX ; MÞand ðY ; NÞ are two minimal vector space. f : X ! Y is linear and m-continuous. If A is m-bounded in X then f(A) is m-bounded in Y. Proof. Suppose V is any m-neighborhood of 0 in Y. From the assumption f1(V) = U is an m-open neighborhood of 0 in X. There exist t0 > 0 such that tA U for each jtj 6 t0. Then tf(A) = f(tA) V . Therefore, f(A) is m-bounded. h Next result introduces other brand of m-continuity which we call it m-quasi continuity. Definition 3.15. f : ðX ; MÞ ! ðY ; NÞ is called m-quasi continuous at x 2 X if for any m-open set V in Y containing f(x), f1(V) can be expressed as a union of elements in M. f is m-quasi continuous if it is m-continuous at each x 2 X. A X is called quasi continuous if for each sequence (xn)n in A; 1n xn ! 0. Proposition 3.16. Suppose T : ðX ; MÞ ! ðY ; NÞ is a linear m-quasi continuous map. Then T is m-bounded (i.e., T sends m-bounded sets to m-bounded sets). Proof. Choose 1 any m-bounded sequence (xn)n in X, we must show that (T(xn)) is m-bounded sequence; i.e., 1 ðTx Þ ¼ T x tends to zero. To do this, suppose V is an m-open neighborhood of 0 in Y. From the assumption n n n n
M. Alimohammady, M. Roohi / Chaos, Solitons and Fractals 33 (2007) 1348–1354
S T 1 V ¼ a2A U a , so there is a0 2 A for which 0 2 Ua0 . There is N > o; 1 T ðxn Þ ¼ T 1n xn 2 TUa0 V which it concludes the proof. h n
1 x n n
1353
2 U a0 for all n P N. Therefore,
Proposition 3.17. Suppose T : ðX ; MÞ ! ðY ; NÞ is a linear map then the following are equivalent: (a) T is m-quasi continuous at 0. (b) T is m-quasi continuous at each point of X.
Proof. (b) N (a) is trivial. To see (a) N (b), assume x 2 X, V is an m-neighborhood of Tx. Then Tx + V is an m-neighborhood of 0, so [ U a. x þ T 1 ðV Þ ¼ T 1 ðTx þ V Þ ¼ a2A
Therefore, T 1 ðV Þ ¼ x þ
[ a2A
Ua ¼
[
x þ U a;
a2A
which implies that T is quasi m-continuous at x.
h
4. Conclusions We did some results on minimal structure as an extended notion of topology. 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. Also, 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 extend the accuracy of the approximation in many fields of applications such as quantum gravity [5,21], discussion making [12], image processing [8], digital topology [7], and others [14,15]. The paper presents an alternative an in fact an extended study of topology called minimal structure for linear spaces. Some of the new conclusions following from the results investigated in paper are: (a) (b) (c) (d)
f + g : X ! Y is m-continuous, if f, g : X ! Y are m-continuous function. f : X ! X defined by f(x) = t0x + x0 is m-continuous. If A is m-open set, then x0 + A is m-open too. A linear map T : ðX ; MÞ ! ðY ; NÞ is m-quasi continuous at 0 if and only if T is m-quasi continuous at each point of X.
References [1] Csaszar A. Generalized topology: generalized continuity. Acta Math Hunger 2002;96:351–7. [2] El Naschie MS. On the uncertainty of Cantorian geometry and two-slit experiment: generalized continuity. Chaos, Solitons & Fractals 1998;9(3):517–29. [3] El Naschie MS. On the unification of heterotic strings, M theory and e(1). Chaos, Solitons & Fractals 2000;11(14):2397–408. [4] El Naschie MS. Non-Euclidean spacetime structure and the two slit experiment. Chaos, Solitons & Fractals 2005;26:1–6. [5] El Naschie MS. Superstrings, knots, and noncommutative geometry in space. Int J Th Phys 1998;37(12):2935–51. [6] Lugojan S. Generalized topology. Stud Cere Math 1982;34:348–60. [7] Kong TY, Kopperman R, Meyer PR. A topological approach to digital topology. Am Math Mon 1991;98:901–17. [8] Kong TY, Kopperman R, Meyer PR. Guest editors preface to special issue on digital topology. Topol Appl 1992;46:173–9. [9] Maki H. On generalizing semi-open sets and preopen sets. In: Meeting on topological spaces theory and its application; August 1996. p. 13–18. [10] Maki H, Umehara J, Noiri T. Every topological space is pre T1/2. Mem Fac Sci Kochi Univ Ser A Math 1996;17:33–42. [11] Noiri T. On Km-sets and related spaces. In: Proc 8th meetings on topological spaces theory and its application; August 2003. p. 31–41.
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Pawlak Z. Raugh sets, raugh relations and raugh functions. Bull Pol Ac Math 1996:15–9. 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. Shuliang W, Deren LI, Wenzhong SHI, Xinzhou W, Deyi LI. Raugh spatial description. IAPRS 2002;44(2):20–3. Stadler BMR, Stadler PF. Generalized topological spaces in evolutionary theory and combinatorial chemistry. J Chem Inf Comput Sci 2002;42:577–85. Sumners DW. New scientific applications of geometry and topology. Proc sympos appl math, vol. 45. Providence, RI: American Mathematical Society; 1994. Swartz C. An introduction to functional analysis. New York: Marcel Dekker; 1992. Wang JC. DNA topoisomerases. Sci Amer 1982;247:94–109. Wang JC. DNA topoisomerases. Ann Rev Biochem 1985;54:665–97. Wasserman SA, Cozzarelli NR. Biochemical topology: applications to DNA recombination and replication. Science 1986;232: 951–60. Witten E. Reflections on the fate of spacetime. Phys Today 1996:24–30.
Linear minimal space M. Alimohammady a
a,*
, M. Roohi
b
Department of Mathematics, University of Mazandaran, Babolsar, Iran b Islamic Azad University, Sari-branch, Iran Accepted 23 January 2006
Abstract This paper deals with minimal linear spaces, m-continuity and m-boundedness. In particular, it is found that in a linear minimal space (X ; M) the assignment x # t0x + x0 from X to X is m-continuous. On the other hand, the convex hull of an m-neighborhood of 0 is an m-neighborhood if (X ; M) has property U. 2006 Elsevier Ltd. All rights reserved.
1. Introduction and preliminaries This paper deals with minimal structure as an extended notion of topology via linear spaces which has many applications in applied sciences such as quantum gravity [5,21], discussion making [12], image processing [8], digital topology [7], and others [14,15]. The topology of quantum spacetime is shadowed closely by the Mobius geometry of quasiFuschian and Kleinian groups and that is the cause behind the phenomena of high-energy particle physics [5]. 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. Minimal structures may have very important applications in quantum particles physics, particularly in connection with string theory and e1 theory [2–4]. One of conclusions of advanced scientific research into the very basic question related to the quintessence of natural science and philosophy is topology and can be minimal structure. For example, topology has many applications in string theory, in the study of DNA replication and recombination, and in areas of statistical mechanics. DNA topology is the focus of a subdiscipline within molecular biology and as a term refers to both the knot-like arrangements that segments of DNA may assume and to the mathematics that pertains to them. The topology of DNA topoisomers is important to replication, transcription and recombination, including the recombination events important to the life cycles of many viruses. Topoisomerases are enzymes that change the topology of DNA; for more details we refer to [16,18–20]. Many properties on topological spaces such as boundedness and continuity are discussed via minimal notion. While others are far from being completely devoted in its foundation. So, this paper is devoted to study the class of minimal structures via linear spaces. Some characterizations and several properties of these concepts are discussed.
*
Corresponding author. E-mail addresses: [email protected] (M. Alimohammady), [email protected] (M. Roohi).
0960-0779/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.01.100
M. Alimohammady, M. Roohi / Chaos, Solitons and Fractals 33 (2007) 1348–1354
1349
In [10] authors introduced minimal structures and minimal spaces. Some results about minimal spaces can be found in [1,6,9,11,13]. As is well known, a topological vector space is a vector space endowed with a topology for which two operations + : X · X ! X ((x,y) # x + y) and : F X ! X ððt; xÞ 7! txÞ are continuous. In what follows we will consider linear minimal vector space, it is shown that x # t0x + x0 is m-continuous, further the convex hull of an m-neighborhood of 0 is an m-neighborhood if (X ; M) has the property U. For easy understanding of the material incorporated in this paper we recall some basic definitions. For details on the following notions we refer to [9,13,17]. Definition 1.1 [10]. A family M PðX Þ is said to be a minimal structure on X if ;; X 2 M. In this case (X ; M) is called a minimal space. Example 1.2 [10]. Let (X, s) be a topological space. Then, M ¼ s; SOðX Þ; POðX Þ; aOðX Þ and bO(X) are minimal structures on X. Definition 1.3 [10]. A set A 2 PðX if A 2 M B 2 PðX Þ is an m-closed set S Þ is said to be an m-open set (m-neighborhood) T if Bc 2 M. We set m IntðAÞ ¼ fU : U A; U 2 Mg and m ClðAÞ ¼ fF : A F ; F c 2 Mg. Proposition 1.4 [10]. 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).
Definition 1.5 [13]. A minimal space (X ; M) enjoys the property U if ‘‘the arbitrary union of m-open sets is m-open’’. (X ; M) has the property I if ‘‘the any finite intersection of m-open sets is m-open’’. Proposition 1.6 [13]. For a minimal structure M on a set X, the following are equivalent: (a) M has the property U. (b) If m Int(A) = A, then A 2 M. (c) If m Cl(B) = B, then Bc 2 M.
2. Minimal product spaces In the following section we introduce minimal structure for product of minimal spaces. Lemma 2.1. The intersection and union of any collection of minimal structures is a minimal structure. Proof. It is straightforward.
h
Definition 2.2. f : ðX ; MÞ ! ðY ; NÞ is said to be minimal continuous (briefly m-continuous) if f1(U) is an m-open set, where U is any m-open set in N. Theorem 2.3. The composition of two m-continuous functions is an m-continuous function. Proof. It is direct.
h
In the next result we would like to introduce product minimal structure. First we will consider the initial minimal structure for a family of functions.
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M. Alimohammady, M. Roohi / Chaos, Solitons and Fractals 33 (2007) 1348–1354
Proposition 2.4. Suppose ðY ; NÞ is a minimal space and f : X ! ðY ; NÞ is a function. Then there is a weakest minimal structure on X for which f is m-continuous. Proof. The required minimal structure is M ¼ f 1 ðNÞ ¼ ff 1 ðV Þ : V 2 Ng.
h
Corollary 2.5. Suppose ðX ; MÞ is a minimal structure and Y X. Then there is a weakest minimal structure on Y say N, such that the inclusion map i : ðY ; NÞ ! ðX ; MÞ is m-continuous. Proof. It is enough to consider i instead of f in Proposition 2.4
h
We call N the induced minimal structure by M on Y. Corollary 2.6. Suppose Y X and f : ðX ; MÞ ! ðZ; PÞ is m-continuous. Then f jY is m-continuous if Y is endowed with induced minimal structure. Proof. That f jY is m-continuous follows from f jY = foi and Theorem 2.3, where i : Y ! X is the inclusion map.
h
Theorem 2.7. Consider a family fðX a ; Na Þ : a 2 Ag of minimal spaces and ffa : X ! ðX a ; Ma Þ : a 2 Ag a family of functions. Then there is a weakest minimal structure M on X such that fa’s are m-continuous. Proof. It is easy to see that M ¼ Na ; a 2 Ag. h
S
a Ma
is the required minimal structure, where Ma ¼ f 1 ðNa Þ ¼ ffa1 ðV Þ : V 2
Theorem 2.8. Consider a family ffa : X ! ðX a ; Ma Þ : a 2 Ag of functions, where ðX a ; Ma Þ are minimal spaces. Equipped X by minimal structure M generated by ffa : a 2 Ag. f : ðY ; NÞ ! ðX ; MÞ is m-continuous if and only if fa f is m-continuous function for all a 2 A. Proof. If f is m-continuous function, then from Theorem 2.3 for each a 2 A; fa f is m-continuous. For the converse, suppose on the contrary fa f’s are m-continuous functions for each a 2 A but f is not m-continuous. Hence, there is B 2 M such that f 1 ðBÞ 62 N. Then one of the following two statements holds: (i) 9a0 2 A; 9Ba0 2 Ma0 such that B ¼ fa01 ðB0 Þ. (ii) 8a 2 A; 8Ba 2 Ma ; B 6¼ f 1 ðBa Þ. 1 ðB0 ÞÞ ¼ ðfa0 f Þ1 ðBa0 Þ, Therefore, ðfa0 f Þ1 ðBa0 Þ 62 N which is a In case (i), we have f 1 ðBÞ ¼ f 1 ðfa0 contradiction, because f is an m-continuous function. In the other case, since f1(;) = ; and f1(Y) = X so, B 62 {;, Y}. Then M n fBg is a minimal structure on X and so for each a 2 A; fa : ðX ; M n fBgÞ ! ðY ; NÞ is m-continuous, which is a contradiction according to the choice of M. h
As a consequence of Theorem 2.7 we present the product minimal structure for an arbitrary family fðX a ; Ma Þ : a 2 Ag of minimal spaces. Product minimal structure on X ¼ Pa2A X a is the weakest minimal structure on X (denoted by M ¼ Pa Ma Þ, such that for each b 2 A the canonical projection pb : Pa2A X a ! X b is m-continuous. Corollary 2.9. For any family fðX a ; Ma Þ : a 2 Ag of minimal spaces, product minimal structure on X ¼ Pa2A X a exists. Proof. Only we must consider pa in Theorem 2.7 instead of fa.
h
Corollary 2.10. Suppose X ¼ Pa2A X a ; where each Xa is a minimal space. Endow X to the product minimal structure generated by fPa X a : a 2 Ag. Then f is m-continuous function if and only if pa f of is m-continuous for all a 2 A, where f : ðY ; NÞ ! ðX ; MÞ is a function. Proof. It follows immediately from Theorem 2.8
h
Corollary 2.11. Suppose f : ðX ; MÞ ! ðY ; NÞ and g : ðX ; MÞ ! ðZ; PÞ are m-continuous functions. Then the function f g : ðX ; MÞ ! ðY Z; N PÞ defined by (f · g)(x) = (f(x), g(x)) is m-continuous.
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Proof. p1(f · g)(x) = p1(f(x), g(x)) = f(x) and p2(f · g)(x) = p2(f(x), g(x)) = g(x). Hence, p1(f · g) and p2(f · g)(x) are mcontinuous. Then Corollary 2.10 implies that f · g is m-continuous. h
3. Linear minimal spaces and m-bounded sets In this section we show that x # t0x + x0 is m-continuous. Definition 3.1. A linear minimal structure on a vector space X over the complex field F is a minimal structure M on X such that the two mappings þ :X X ! X ; ðx; yÞ 7! x þ y :F X ! X ; ðt; xÞ 7! tx are m-continuous, when F has the usual topology and F X and X · X the corresponding product minimal structures. A linear minimal space (or m-linear space) is a vector space together with a linear minimal structure. Proposition 3.2. Suppose f, g : X ! Y are m-continuous functions, where X and Y are minimal space and linear minimal space, respectively. Then f + g : X ! Y is m-continuous function too. Proof. Consider h : Y · Y ! Y which is defined by h(y1, y2) = y1 + y2. That f + g is m-continuous follows from Theorem 2.8, f + g = ho(f · g) and Corollary 2.11. h Theorem 3.3. Let X be a linear minimal space, t0 2 F and x0 2 X. Then f : X ! X defined by f(x) = t0x + x0 is m-continuous. If t0 5 0, then f is a minimal homeomorphism. Proof. Consider f1 : X ! F and f2 : X ! X defined by f1(x) = t0 and f2(x) = x, respectively. It is easy to see that f1 and f2 are m-continuous. Consequently, Corollary 2.10 implies that f1 f2 : X ! F X is m-continuous. Since g : F X ! X defined by g(t, x) = tx is m-continuous, so g (f1 · f2), and f1 · f2 are m-continuous. Then from Proposition 3.2, f = g (f1 · f2) + x0 is m-continuous. For the second assertion only we must note that f 1 ðxÞ ¼ t10 ðx x0 Þ. h Corollary 3.4. If A is an m-open set in a linear minimal space X, and t 5 0, then tA is m-open. Proof. Both tA = f1(A) and m-continuity of f : X ! X defined by f ðxÞ ¼ 1t x complete the proof.
h
Corollary 3.5. Let (X ; M) be a minimal structure, A be an m-open set in X and x0 2 X. Then x0 + A is an m-open set. Proof. Consider g = h (f · idX), where f : ðX ; MÞ ! ðX ; MÞ is defined by f(x) = x0 and h : X · X ! X is defined by h(x, y) = x + y. Similar to the proof of Theorem 3.3, f · idX is m-continuous. Hence, g is m-continuous by Theorem 2.8. That x0 + A is m-open follows from this and the fact that g1(A) = x0 + A. h In the following we deal with m-bounded sets in a linear minimal space. Definition 3.6. A X is called balanced if D Æ A = A. We call D Æ A the balanced hall of A and is denoted by bal(A). Proposition 3.7. Suppose (X ; M) is a fuzzy minimal space with the property U. Then the balanced hall of any m-neighborhood of 0 is also an m-neighborhood of 0. Proof. Since balðU Þ ¼ D U ¼
[
tU
jtj61
and from Theorem 3.3 each t Æ U is m-open, so property U implies that bal(U) is m-open.
h
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Proposition 3.8. m Cl(A) is balanced if A is balanced. Proof. Consider f : F X ! X by f(a, x) = a Æ x is m-continuous, so f(m Cl(D · A)) m Cl(f(D · A)) and so f(D · m Cl(A)) m Cl(f(D · A)). Then D Æ (m Cl(A)) m Cl(D Æ A) = m Cl(A) which implies that m Cl(A) is balanced. h Proposition 3.9. Suppose U is an m-(open)neighborhood of 0 in X and V is the family of m-balanced halls of members of U. Then V is an m-open neighborhood base at 0. Proof. Assume U 2 U. Since 0 + 0 = 0 so there are a P 0 and m-open neighborhood W1 of 0 such that (aD) Æ W1 U. Therefore, W2 = D Æ (aW1) U, so W2 is an m-open balanced neighborhood contained in U. h Definition 3.10. In a minimal vector space ðX ; MÞ; A X is said to be (a) m-bounded set if there is t0 > 0 such that A tU, where U is any m-open set containing 0 and jtj 6 t0, (b) convex if [ t1 V þ þ tn V coðV Þ ¼ Pn 06ti 61;
i¼1
ti ¼1;n2N
is a subset of A.
Proposition 3.11. If A B and B is m-bounded, then A is m-bounded too. Proof. It is straightforward.
h
Proposition 3.12. Suppose A and B are m-bounded, then A [ B is m-bounded too. Proof. Consider U as an m-neighborhood of 0. Prom the assumption there are t1, t2 > 0 such that t 6 min{t1, t2} implies that tA U and tB U. Then t(A [ B) = tA [ tB U. h Proposition 3.13. Suppose (X ; M) has property I, A and B are m-bounded. Then A + B is m-bounded. Proof. Suppose W is any m-open set containing 0, from m-continuity of + there are two m-neighborhoods U1 and U2 of 0 for which U1 + U2 W. Set U1 \ U2 = U. Then U + U W. There are t1,t2 > 0 such that jtj 6 min{t1,t2} implies that tA U and tB U. Then t(A + B) = tA + tB U + U W. h The following result shows the stability of m-bounded sets under m-continuous functions. Proposition 3.14. Suppose ðX ; MÞand ðY ; NÞ are two minimal vector space. f : X ! Y is linear and m-continuous. If A is m-bounded in X then f(A) is m-bounded in Y. Proof. Suppose V is any m-neighborhood of 0 in Y. From the assumption f1(V) = U is an m-open neighborhood of 0 in X. There exist t0 > 0 such that tA U for each jtj 6 t0. Then tf(A) = f(tA) V . Therefore, f(A) is m-bounded. h Next result introduces other brand of m-continuity which we call it m-quasi continuity. Definition 3.15. f : ðX ; MÞ ! ðY ; NÞ is called m-quasi continuous at x 2 X if for any m-open set V in Y containing f(x), f1(V) can be expressed as a union of elements in M. f is m-quasi continuous if it is m-continuous at each x 2 X. A X is called quasi continuous if for each sequence (xn)n in A; 1n xn ! 0. Proposition 3.16. Suppose T : ðX ; MÞ ! ðY ; NÞ is a linear m-quasi continuous map. Then T is m-bounded (i.e., T sends m-bounded sets to m-bounded sets). Proof. Choose 1 any m-bounded sequence (xn)n in X, we must show that (T(xn)) is m-bounded sequence; i.e., 1 ðTx Þ ¼ T x tends to zero. To do this, suppose V is an m-open neighborhood of 0 in Y. From the assumption n n n n
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S T 1 V ¼ a2A U a , so there is a0 2 A for which 0 2 Ua0 . There is N > o; 1 T ðxn Þ ¼ T 1n xn 2 TUa0 V which it concludes the proof. h n
1 x n n
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2 U a0 for all n P N. Therefore,
Proposition 3.17. Suppose T : ðX ; MÞ ! ðY ; NÞ is a linear map then the following are equivalent: (a) T is m-quasi continuous at 0. (b) T is m-quasi continuous at each point of X.
Proof. (b) N (a) is trivial. To see (a) N (b), assume x 2 X, V is an m-neighborhood of Tx. Then Tx + V is an m-neighborhood of 0, so [ U a. x þ T 1 ðV Þ ¼ T 1 ðTx þ V Þ ¼ a2A
Therefore, T 1 ðV Þ ¼ x þ
[ a2A
Ua ¼
[
x þ U a;
a2A
which implies that T is quasi m-continuous at x.
h
4. Conclusions We did some results on minimal structure as an extended notion of topology. 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. Also, 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 extend the accuracy of the approximation in many fields of applications such as quantum gravity [5,21], discussion making [12], image processing [8], digital topology [7], and others [14,15]. The paper presents an alternative an in fact an extended study of topology called minimal structure for linear spaces. Some of the new conclusions following from the results investigated in paper are: (a) (b) (c) (d)
f + g : X ! Y is m-continuous, if f, g : X ! Y are m-continuous function. f : X ! X defined by f(x) = t0x + x0 is m-continuous. If A is m-open set, then x0 + A is m-open too. A linear map T : ðX ; MÞ ! ðY ; NÞ is m-quasi continuous at 0 if and only if T is m-quasi continuous at each point of X.
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