On Bicomplex Nets and their Confinements

Transcription

1 American Journal of Mathematics and Statistics 0; (): 8-6 DOI: 0 593/jajms0000 On Bicomplex Nets and their Confinements Rajiv K Srivastava *, S Singh Department of Mathematics, Institute of Basic Science, Khandari Campus, Dr B R Ambedar University, Agra 8 00, India Abstract We have initiated the study of nets with bicomplex entries Due to the multi dimensionality of the bicomplex space there arise different types of tendencies called confinements The bicomplex space equipped with real order topology as well as idempotent order topology exhibits interesting and challenging behaviour of nets Different types of confinements have been characterized in terms of convergence of the component nets In the final section, certain relations between bicomplex nets and their projection nets have been derived Keywords Bicomplex Numbers, Order Topology, Bicomplex Net, Real Confinement, Idempotent Confinement, Projection Net Introduction The symbols C 0, C and C denote sets of real numbers, complex numbers and bicomplex numbers, respectively A bicomplex number is defined as (cf[,]) ξ = x ix ix 3 ii x 4, where x p C0, p 4, z, z C, i = i = and i i = i i With usual binary compositions, C becomes a commutative algebra with identity Besides the additive and multiplicative identities 0 and, there exist exactly two non-trivial idempotent elements denoted by e and e defined as e = ( i i ) / and e = ( i i ) / Note that e e = and e e = 0 A bicomplex number ξ = z i z can be uniquely expressed as a complex combination of e and e as (cf [3]) ξ= z iz = ( z iz ) e ( z iz ) e = ξ e ξ e, where ξ= z iz and ξ= z iz The complex coefficients ξ and ξ are called the idempotent components and the combination ξ e ξ e is nown as idempotent representation of bicomplex number ξ The auxiliary complex spaces A and A are defined * Corresponding author: ssuhdev09@yahoocom ( S Singh ) Published online at Copyright 0 Scientific & Academic Publishing All Rights Reserved as follows: A = { z iz ; z,z C} = { ξ : ξ C } and A= { z iz ; z,z C } = { ξ: ξ C} The idempotent representation ( z i z ) e ( z i z ) e = ξ e ξ e associates with each point ξ = z i z in C, the points ξ = z iz and ξ = z iz in A and A, respectively and to each pair of points ( z, w) A A, there corresponds a unique bicomplex point ξ = z e w e Some updated details of the theory of Bicomplex Numbers can be found in [5,6] Order Topologies on C Srivastava[3] initiated the topological study of C He defined three topologies on C, viz, norm topology τ, complex topology τ and idempotent topology τ 3 and has proved some results on these topological structures Throughout, < denotes the ordering of real numbers and denotes the dictionary ordering of the complex numbers Denote by R, the dictionary ordering of bicomplex numbers expressed in the real component form The order topologies induced by this ordering will be called as Real Order Topology (cf[4]), denoted by τ 4 and is generated by the basis comprising of the members of the following families of subsets of C : x ix ix 3 iix,y 4 J = :x< y iy iy 3 iiy 4 R a ix ix 3 iix,a 4 J = :x< y iy iy 3 iiy 4 R

2 American Journal of Mathematics and Statistics 0; (): a ib ix 3 iix,a 4 J 3= :x3< y3 ib iy 3 iiy 4 R a ib ic iix 4,a, J 4= :x4< y4 ib ic ii y4 R the set ( ξ, η denoting the open interval with respect ) R to the ordering R Throughout the discussion, we shall consider some special types of subsets of the bicomplex space C, equipped with real order topology A set of the type { ξ : ξ = x i x i x 3 i i x 4 : a < x and x < b } is called an open space segment A set of the type {ξ: ξ = a i x i x 3 i i x 4 } is called a frame and is denoted as ( x = a ) A set of the type {ξ: ξ = a i x i x 3 i i x 4 ; b < x and x < c} is called an open frame segment A set of the type {ξ: ξ = a i b i x 3 i i x 4 } is called as plane and is denoted as ( = a, x b ) A set of the x = : < type { ξ = a i b i x i i x ; c < x and x d} ξ is called an open plane segment A set of the type {ξ: ξ = a i b i c i i x 4 } is called as a line and is denoted as ( = a, x = b, x c ) A set of x 3 = b i c i i x 4 ; d x the type { ξ : ξ= a i < 4 and x 4 < e } is called an open line segment Note that J is a family of open space segments, J is a family of open frame segments, J 3 is a family of open plane segments and J 4 is a family of open line segments Further, we shall consider some special types of subsets of the bicomplex space C equipped with the idempotent order topology, (cf[4]) Denote by, the dictionary ordering of the bicomplex numbers expressed in the idempotent form The order topology induced by this ordering is called as Idempotent Order Topology (cf[4]) Hence, idempotent order topology, τ 6 is generated by the basis B 6 comprising of members of the following families of subsets of C : L = ξ e ξ e, ηe ηe ξ η {( ) } {( ξ e ξ e, ξ e ηe ) ξ η } : L = :, the set ( ξ, η) denoting the open interval with respect to the ordering A set of the type { ξ : a < Re ξ < b} is called an open space segment ξ : Re ξ = a is called an frame A set of the type { } and is denoted as ( Re a) ξ = { : Re ξ = a, b < Im ξ < c} A set of the type ξ is called an open frame segment ξ : Re ξ = a, Im ξ = b is called as A set of the type { } an plane and is denoted as ( Re ξ= a, Im ξ= b) A set of the type { ξ : Re ξ = a, Im ξ = b, c < Re ξ < d} is called an open plane segment A set of the type { ξ : Re ξ = a, Im ξ = b, Re ξ = c} is called an line and is denoted as ( Re ξ = a, Im ξ = b, Re ξ = c) A set of the type { ξ : Re ξ = a, Im ξ = b, Re ξ = c, d < Im ξ < e} is called an open line segment Remar Note that, L and L can also be described as L = N N and L = N3 N 4, where { } N = ξ e ξe, η e ηe : Re ξ < Re η ( ξ e ξe, η e ηe) N = :Re ξ = Re η, Im ξ< Im η N 3 = ( ξe ξe, ξe ηe ) : Re ξ < Re η ( ξ e ξe, ξ e ηe) N 4 = : Re ξ= Re η, Im ξ< Im η { } In other words, B 4 = J p and B = L = p 4 = 4 6 p N p p = p = Remar The geometry of the Cartesian idempotent set determined by A and A, ie, A e A is entirely different from the geometry of Cartesian set determined by C and C, ie, C c C (for definitions see[3]) Obviously, the members of the families N, N, N 3 and N 4 are open space segments, open frame segments, open plane segments and open line segments, respectively 3 Confinements of Bicomplex Nets in the Real Order Topology 3 Static and Eventually Static Bicomplex Net Let D be an arbitrary directed set Then a bicomplex net can be defined as Φ : D C such that D Φ = ξ = x ix i x ii x z i = z = ξ e ξ e Further, a bicomplex net { } is said to be static on ξ if = ξ, D It is said to be eventually static on ξ if there exists β D such that = ξ, β

3 0 R K Srivastava et al: On Bicomplex Nets and their Confinements 3 Real Frame Confinement (RF Confinement) { ξ } = { x ix i x ii x } is said to be Real Frame confined (in short, RF confined) to the frame ( x = a ), if it eventually in every member of the family J (of open space segments) containing the frame ( x = a ) 33 Real Plane Confinement (RP Confinement) { ξ } = { x ix i x ii x } is said to be Real Plane confined (in short, RP confined) = a, x b, if it is eventually in every to the plane x = member of the family J (of open frame segments) con- = a, x b taining the plane x = 34 Real Line Confinement (RL Confinement) { ξ } = { x ix ix ii x } is said to be Real Line confined (in short, RL confined) to = a, x = b, x c, if it is eventually in the line x 3 = every member of the family J 3 (of open plane segments) = a, x = b, x c containing the line x 3 = 35 Real Point Confinement (R Point Confinement) { ξ } = { x ix ix ii x } is said to be Real Point confined (in short, R Point confined) to the bicomplex point ξ = i b i c i i d, if a it is eventually in every member of the family J 4 (of open line segments) containing the point ξ 36 Remar Note that if a bicomplex net { } is RF confined to the frame ( x = a), it will not be eventually in any member of the family J (and therefore will not be RP confined to any plane) unless { x } is eventually static on a (and in this case, { ξ} will be RP confined to the plane ( x = a, x = b ) provided that { x } b ) Similar cases will arise with the other types of bicomplex nets in the real form 37 Remar is a necessary but not sufficient condition for the convergence of the net in the classical sense (ie, in the topology τ induced by the Euclidean norm) In fact, every eventually static net { x }, 4, converges and therefore, R ξ to ξ implies convergence of The R Point confinement of a bicomplex net { ξ} Point confinement of { } { } the bicomplex net { } ξ to ξ in the norm To verify insufficiency, consider ξ on the directed set ( Q ), of positive rationals (with usual ordering) defined as follows: ξ = x ix ix ii x, where x = /, 4 The net converges to the point ξ = i i 3 ii 4 in the norm but not R Point confined to ξ As a matter of fact, no member of { } belongs to the line segment i i 3 i i 4, i i 3 i i ( ( 4 )) R 38 Theorem { ξ } = { x ix i x ii x } is RF confined to the frame ( x = a ) if and only if the net { x } converges to a Proof: Assume that { } let x converges to a Given > 0, = { ξ : ξ = x ix ix 3 ii x 4 ; a < x < a } be the member of J containing the frame ( x = a) Since, { x } converges to a, an index D B (3) β such that x p, yp C0, p 3,, x ( a, a ), β a < x < a, β ( a ) ix i x 3 ii x 4 R x ix ix ii x and x ix i x ii x R ( a ) iy iy3 ii y4 x ix ix ii x ( a i x i x 3 i i x 4, a i y i y 3 i i y 4 ) R x p, yp C0, p 3 and β So that the net { ξ } = { x ix ix ii x } is eventually in B Since > 0 is arbitrary and every member of J contains a B (for some > 0 ), { ξ} is RF confined to the frame ( x = a) Conversely, let the bicomplex net { ξ } = { x ix i x ii x } be RF confined to the frame ( x = a) Therefore, the bicomplex net { ξ } = { x ix i x ii x } is eventually in every member of J containing the frame ( x = a) B ( > 0 ) defined by (3) Thus, β D such that ξ = x ix ix ii x B, β a < x < a, β In particular, { } is eventually in { } a x

4 American Journal of Mathematics and Statistics 0; (): 8-6 Hence the theorem 39 Theorem = { ξ } { x ix ix ii x } is RP confined to the plane ( x = a, x = b) if and only if the net { x } is eventually static on a and net { x } converges to b Proof: Let the net { x } the net { } be eventually static on a and x converge to b Since { x } is eventually static on a, β D, such that β, x = a Given > 0, let { ξ : ξ = a i x i x3 ii x 4 ; b < x < b } J containing the plane ( a, x b) I = (3) be a member of x = = As the net { x } converges to b, there exists some γ D such that x b,b, γ Since β, γ D, there exists some δ D such that δ β and δ γ Therefore, x = a and x ( b,b ), δ b < x < b, δ x i x i x ii x ( a i ( b ) i x 3 i i x 4, a i ( b ) i y 3 i i y 4 ) R for any x 3, x 4, y3, y4 C0, δ So that the bicomplex net { ξ } = { x ix i x ii x } is eventually in I Since > 0 is arbitrary and every member of J contains an I for some > 0, the bicomplex net { } is RP confined to the plane ( x = a, x = b ) Conversely, suppose that the bicomplex net { ξ } = { x ix ix ii x } is RP confined to the plane ( x = a, x = b ) Therefore, it is eventually in every member of the type I ( > 0 ) defined by (3), of the family J containing the plane ( x = a, x = b ) So that for given > 0, there exists some β D such that { ξ } I, β x = a and x ( b, b ), β x is eventually static on a and Therefore, the net { } the net { } x converges to b On the similar lines, the following theorems can be proved 30 Theorem { } = { x i x i x i i } ξ x is RL confined to the line ( x a, x = b, x3 = c) only if the nets { x } and { x } a and b, respectively and the net { } 3 Theorem = if and are eventually static on x converges to c { } = { x i x i x i i } ξ x is R Point confined to the bicomplex point i b i c i i d x, a if and only if the nets { x }, { } and { x } are eventually static on a, b and c, respectively and the net { } 3 Theorem x is converges to d (i) Every R Point confined bicomplex net is RL confined (ii) Every RL confined bicomplex net is RP confined (iii) Every RP confined bicomplex net is RF confined The converses of these implications are not true, in general Proof: In fact, if a bicomplex net { ξ} confined to the bicomplex point a x a, x = b, x3 = is R Point i b i c i i d, then it is RL confined to the line ( c) = Similarly, a bicomplex net which is RL confined to the line = a, x = b, x c is RP confined to the plane ( x 3 = ) ( x = a, x = b ) and a bicomplex net which is RP confined to the plane ( x = a, x = b ) is RF confined to the frame ( x = a) That the converse is not true, in general, is shown with the help of the two examples below 33 Example Consider the directed set ( Q, ) net { ξ} as follows: Define the bicomplex { } = { x i x i x i i } ξ x such that x = a x, x =, x 3 = and x =, where the net { } Q, Therefore, the net { } / x is eventually static on 0 x is eventually static on a and the net { } { } is RP confined to the plane ( x a, x = b) the net { } x converges on 0 So that the bicomplex net = Since, x is eventually static on a Therefore, the bicomplex net { ξ} family is eventually in every member of the J containing the frame ( x = a) ξ is RF confined to the frame ( a) Hence, the bicomplex net { } 34 Example Consider the bicomplex net = x i x { } { i x i i } ξ x x =

5 R K Srivastava et al: On Bicomplex Nets and their Confinements such that x = a ( ), x =, x 3 = / and x 4 = 3, Q This bicomplex net is RF confined to the frame ( x = a) The component net { x } converges to a but component net { x } is not convergent Therefore, the bicomplex net { ξ } = { x ix ix ii x } is not RP confined to any plane contained in the frame ( x = a) 4 Confinements of Bicomplex Nets in Idempotent Order Topology In this section, we assume C to be furnished with the idempotent order topology (cf [4]) Hence, the net = x i x i x i i { } { } ξ x = ξe ξe, will be treated as the net { } { } where ξ e ξ e = ( x x) i( x x) e (4) ( x x) i( x x) e For the sae of brevity, we shall express the numbers the numbers x x 4 and x x 3 as Re ξ and Im ξ, respectively Thus ( Re a) ξ = denote the frame x x 4 = a, whereas { Re } denote the net { x x } and so on Under these notations, (4) can be rewritten as ξ e ξ e = ( Re ξ iim ξ) e (4) Re ξ i Im ξ e ( ) Note that for the net { } both the nets { } nets { x } and { } x and { x } Re to be convergent, either are convergent or both x are divergent but { } convergent Similarly, for net { } if either both the nets { } or both nets { } { } { Re } and { } x and { } x and { x } Re is Im will be convergent x are convergent are divergent but Im is convergent The convergence of nets Im can be similarly interpreted 4 Frame Confinement ( F Confinement) { } = { ξ e ξ e } is said to be Frame confined (in short, F confined) to the if it is eventually in every member of frame ( Re ) ξ = a the family N (of open space segments) that contains the frame ( Re ) ξ = a 4 Plane Confinement ( P Confinement) { } = { ξ e ξ e } is said to be Plane confined (in short, P confined) to the plane ( Re ξ = a, Im ξ = b) if it is eventually in every member of the family N (of open frame segments) that contains the plane ( Re = a, Im ξ = b) ξ 43 Line Confinement ( L Confinement) { } = { ξ e ξ e } line is said to be Line confined (in short, L confined) to the Re ξ = a, Im ξ = b, Re ξ = c if it is eventually in every member of the family N 3 (of open plane segments) containing the line Re ξ = a, Im ξ = b, Re ξ = c 44 Point Confinement { } = { ξ e ξ e } ξ is said to ξ be Point confined to the point ξ = e ξe if it is eventually in every member of the family N 4 (of line segments) containing the point ξ = ξe ξe 45 Remar Note that if a bicomplex net { } defined by (4) is F confined to the frame ( Re ξ = a), it cannot be even- N unless { Re } tually in any member of the family is eventually static on a Similar cases will arise with the other types of the confinements of the bicomplex nets with respect to the idempotent order topology 46 Theorem { } frame ( Re = a) Re converges to a ξ is F confined to the ξ if and only if the net { } Proof: Assume that { } Given > 0, let S = { ξ : a < Re ξ < a } Re converges to a (43) be a member of N containing the frame ( Re ) ξ = a Since { Re } a, there exists an index β D such that a < Re ξ < a, β Hence, by the definition of, we have β and x, y, C x < (Re ξ) i(im ξ) ξ ) i(im ξ) < a iy i Im ξ e Re ξ i a i and (Re Re ξ Im ξ ie, ( e (( a ix ) e ( x ix ) e,( a iy ) e ( y iy ) e ) x 3, x 4, y3 and y 4 C, β for any 0 So that the bicomplex net { } is eventually in S Since > 0 is arbitrary and every member of N

6 American Journal of Mathematics and Statistics 0; (): contains S for some > 0, by definition, { } F confined to the frame ( Re = a) ξ ξ = ξ is Conversely, let the bicomplex net { } { ξ e ξ e } be F confined to the frame ( Re ξ = a) By definition, it is eventually in every member of B 6 (in fact, of Re ξ = a In particular, it is eventually in S, for every, defined by (43) Thus, β D such that S, β, ie, such that x, x 3, x 4, y, y3, y4 C0 and β ( a ix ) e ( x3 ix 4) e, ξ ( a ix ) e ( x3 ix 4) e By definition of N and S, we infer N ) containing the frame a < Re ξ < a Re, β ( a, a ) { Re ξ } a Hence the theorem 47 Theorem { } ξ is P confined to the plane ( Re ξ = a, Im ξ = b) if and only if the net { Re } is eventually static on a and the net { Im } converges to b Proof: Suppose that the net { } static at a and the net { } Since the net { } Re is eventually Im converges to b Re is eventually static on a, β D, such that β, Re = a Given > 0, let F = { ξ : Re ξ = a, b < Im ξ < b } (44) be a member of N containing the plane ( Re ξ = a, Im ξ = b) Since the net { } Im converges to b, there exists some γ D such that Im ( b, b ), γ Since β, γ D Then, there exists some δ D such that δ β and δ γ Therefore, Re = a Im, δ ξ and ( b, b ) ( ( )) ( ) ( ( )) ( ) a i b e x3 ix 4 e, ξ a i b e x3 ix 4 e x, x, y, y, δ C0 Therefore, the net { } ξ is eventually in F Since > 0 is arbitrary and every member of N contains an F for some > 0, by the definition, the bicomplex net { } ( Re = a,im ξ= b) ξ ξ is P confined to the plane Conversely, suppose that the bicomplex net { } P confined to the plane ( Re = a,im ξ= b) Therefore, the bicomplex net { ξ} ξ ξ is is eventually in every member of the family N containing the plane ( Re ξ = a,im ξ= b) In particular, the net is eventually in every open frame segment F, ( > 0 ), defined by (44) containing the plane ( Re = a,im ξ= b) ξ Given > 0, there exists some β D such that ξ F, β { } Therefore, by definition of F, x = a and x b, b, β Hence the theorem On similar lines, the following theorems can be proved 48 Theorem { } ξ is L confined to the line ( Re = a, Im ξ = b, Re ξ = c) { Re } and { } respectively and { } 49 Theorem ξ if the nets Im are eventually static on a and b, { } Re converges to c point = ( a i b) e ( c i d) e ξ is Point confined to the ξ if the nets { } { Im } and { } respectively and the net { Im } converges to d 40 Theorem Re, Re are eventually static on a, b and c, (i) Every Point confined bicomplex net is L confined (ii) Every L confined bicomplex net P confined (iii) Every P confined bicomplex net F confined The converses of these implications are not true, in general Proof: In fact, if a bicomplex net { } confined to the point ξ = ( a i b) e ( c i d) e ξ is Point, say, then it is L confined to the line ( Re ξ = a, Im ξ = b, Re ξ = c) Further, a bicomplex net which is L confined to the line Re ξ = a, Im ξ = b, Re ξ = c is P confined to the plane ( Re ξ = a, Im ξ = b) Furthermore, a bicomplex net which is P confined to the plane Re ξ = a, Im ξ = b is F confined to the frame ( Re ξ = a) To show that that the converse of these implications are not true, in general, we give below, in particular, an example of F confined net which is also P confined and

7 4 R K Srivastava et al: On Bicomplex Nets and their Confinements an example of an F confined net which is not P confined 4 Example net Consider the directed set ( Q, ) { } Define the bicomplex 3 a x i (a / ) ξ = 3 Q, i( a / ) ii(a x ) x is eventually static on 0 where { } By (4), the net { } Re is eventually static on a and hence converges to a, the bicomplex net { } is eventually in every member of the family N containing the frame ( Re ξ = to the frame ( Re ξ = a) Also, the net { } net { } Hence the net is F confined Re is eventually static on a and the Im converges to a Therefore, bicomplex net { } the plane ( Re = a, Im ξ= b) 4 Example ξ ξ is P confined to Consider the bicomplex net ( a x ) i ( / ) { ξ } = i( / ) ii( a x ) Q By (4), the net { } and the net { } Re is eventually static on a Im converges to 0 Therefore, the bicomplex net { } is F confined to the frame ( Re ξ = a) Note that although the component net { } Re converges to a, it is not eventually static on a Therefore, the bicomplex net { } is not P confined to any Re ξ = a plane contained in the frame 5 Bicomplex Net and its Projection Nets This section is devoted to the study of relations between confinements of a bicomplex net and the convergence of its projection nets (cf[7]) in the idempotent product topology (cf[4]) Recall the definitions of the auxiliary complex spaces A and A 5 Theorem { } converges to a bicomplex point ξ in the idempotent product topology if and only if the net { } in Proof: If { } ξ A is confined to in A ( =, ) ξ converges to the bicomplex point ξ, it is eventually in every neighbourhood of ξ with respect to the idempotent product topology Note that π ( ξ ) = { } and the projections π : A A A are continuous, =, (cf [5]) e Therefore, the net { } in neighbourhood of π ( ξ), =, Hence the net { } in A, =, The converse is straightforward A is eventually in every A is confined to ξ 5 Note The analogue of the above result is not true for any type of confinement (except Point confinement) of the bicomplex nets with respect to the idempotent order topology on C Further, there is a characteristic difference between the convergence in the idempotent product topology and the confinement in the idempotent order topology in the sense that for any type of confinement (except Point confinement) of a bicomplex net with respect to the idempotent order topology it is not necessary to have all the component nets to be convergent We prove the following results in this context 53 Theorem If the bicomplex net { } frame ( Re = a) confined to the line in ξ is F confined to the ξ, the projection net { } x = a in A Proof: Suppose that the bicomplex net { ξ} in A is defined by Re ξ = a (4), is F confined to the frame Therefore the bicomplex net { } is eventually in every member of the family N (of open space segments) containing the frame ( Re ξ = a) Now, the projection of every member of N on A is a plane segment in A and therefore is a basis element of the dictionary order topology on A Hence, the projection net { } is eventually in every basis element of the dictionary order topology on A containing the line segment x = a in A Hence the net { } x = a in A 54 Theorem If the bicomplex net { } is confined to the line segment ξ is P confined to the plane ( Re ξ = a, Im ξ = b), the projection net { } is confined to the point a ib in A Proof: Suppose that the bicomplex net { ξ} defined by (4), is P confined to the plane Re ξ= a, Im ξ= b

8 American Journal of Mathematics and Statistics 0; (): Therefore, the bicomplex net { } is eventually in every member of the family N (of open frame segments) containing the plane ( Re = a, Im ξ = b) ξ Note that the projection of every member of N on the auxiliary space A is a basis element (in fact, an open interval) of the dictionary order topology on A containing the point i b a So the projection net { } is eventually in every basis element of dictionary order topology on the auxiliary complex space A containing the point a ib Therefore, the projection net { } point a 55 Theorem i b in A If the bicomplex net { ξ} line ( Re ξ = a, Im ξ = b, Re ξ = c) net { } converges to the point i b projection net { } A Proof: Let the bicomplex net { ξ} is confined to the is L confined to the a is confined to the line, the projection in A and the x = c in defined by (4), be L confined to the line Re ξ = a, Im ξ = b, Re ξ = c Therefore, the bicomplex net { ξ} is eventually in every member of the family N 3 (of open plane segments) containing the line Re ξ = a, Im ξ = b, Re ξ = c Since the projection of every member of N 3 on A is a point i b a Therefore, the projection net { } in A is even- A, so it is convergent to the tually static on a ib in point a ib in A The projection on A of every member of N 3 (of open plane segments) is a plane segment in A, which is a basis element of the dictionary order topology on A Therefore, the projection net { } in A is eventually in every basis element of the dictionary order topology on A containing the line x = c in A Hence, the projection net { } the line 56 Theorem x = c in A If the bicomplex net { ξ} bicomplex point ( a i b) e ( c i d) e { } in A is confined to is Point confined to the, the projection net in A converge to the point a ib in A and the projection net { } in A converge to the point c id in A Proof: Let the bicomplex net { ξ} defined by (4), be Point confined to the bicomplex point ( a i b) e ( c i d) e Therefore, it is eventually in every member of the family N (of open line segments) containing the point 4 ( a i b) e ( c i d) e So that the projection net { } in A is confined to the point projection net { } i d in A c 57 Example Consider the bicomplex net { } i b in A and the a in A is confined to the point ξ of Example 4 The net { Re } converges to a and the net { Im } converges to a However, the nets { Re } and { Im } are not convergent and hence the projection net { } in A is not confined in A So the bi- ξ does not converge in the idempotent complex net { } product topology But the projection net { } point a i a in in A is confined to the A Hence the bicomplex net is P confined to the plane ( Re = a, Im ξ = a) 58 Example Consider the directed net ( Q ), net { } ξ Define a bicomplex ξ as follows: ( a ) i ( b h / ) Q, { ξ } = i( b h / ) ii being eventually static on 0 { } h and { } The projection net { } in A is confined to the A and the projection net { } point a i b in in A is confined to the point a i 0 in A Therefore, the bicomplex net is Point confined to the point ( a i b) e ( a i 0) e Since, all the component nets are convergent Therefore, the bicomplex net converges to the point a i b e a i 0 in the classical sense ( ) ( ) e REFERENCES [] Price, G B, 'An introduction to multicomplex spaces and functions', Marcel Deer Inc, New Yor, (99) [] Srivastava, Rajiv K, 'Bicomplex numbers: analysis and applications', Math Student, 7 ( 4) (003), [3] Srivastava, Rajiv K, Certain topological aspects of bicomplex space, Bull Pure & Appl Math, (008), 34

9 6 R K Srivastava et al: On Bicomplex Nets and their Confinements [4] Srivastava, Rajiv K & Singh, Suhdev, 'Certain bicomplex dictionary order topologies', International J of Math Sci & Engg Appls (4), III (00), [5] Srivastava, Rajiv K, Meromorphic Functions of a Bicomplex Variable, International Conference of Special Functions and their Applications, (0) [6] Srivastava, Rajiv K, On the modified consistency in the bicomplex algebra, Symposium Analysis: Some Recent Facets 98 th Indian Science Congress, (0) [7] Willard, S, General topology, Addison Wesley Massachusetts, (970)