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 Applied Mathematics, 2010, 1, 377-386 doi:10.4236/am.2010.15050 Published Online November 2010 (http://www.SciRP.org/journal/am) Copyright © 2010 SciRes. AM Continuous Maps on Digital Simple Closed Curves Laurence Boxer1,2 1Department of Computer and Information Sciences, Niagara University, New York, USA 2Department of Computer Science and Engineering, State University of New York at Buffalo, Buffalo, USA E-mail: boxer@niagara.edu Received August 2, 2010; revised September 14, 2010; accepted September 16, 2010 Abstract We give digital analogues of classical theorems of topology for continuous functions defined on spheres, for digital simple closed curves. In particular, we show the following: 1) A digital simple closed curve S of more than 4 points is not contractible, i.e., its identity map is not nullhomotopic in S; 2) Let X and Y be digital simple closed curves, each symmetric with respect to the origin, such that 5|>| Y (where || Y is the number of points in Y). Let YXf : be a digitally continuous antipodal map. Then f is not nullho- motopic in Y; 3) Let S be a digital simple closed curve that is symmetric with respect to the origin. Let ZSf : be a digitally continuous map. Then there is a pair of antipodes Sxx  },{ such that 1|)()(| xfxf . Keywords: Digital Image, Digital Topology, Homotopy, Antipodal Point 1. Introduction A digital image is a set X of lattice points that model a “continuous object” Y, where Y is a subset of a Eucli- dean space. Digital topology is concerned with deve- loping a mathematical theory of such discrete objects so that, as much as possible, digital images have topological properties that mirror those of the Euclidean objects they model; however, in digital topology we view a digital image as a graph, rather than, e.g., a metric space, as the latter would, for a finite digital image, result in a discrete topological space. Therefore, the reader is reminded that in digital topology, our “nearness” notion is the graphical notion of adjacency, rather than a neighborhood system as in classical topology; usually, we use one of the na- tural lc-adjacencies (see Section 2). Early papers in the field, e.g., [1-8], noted that this notion of nearness allows us to express notions borrowed from classical topology, e.g., connectedness, continuous function, homotopy, and fundamental group, such that these often mirror their analogs with respect to Euclidean objects modeled by respective digital images. Applications of digital topo- logy have been found shape description and in image processing operations such as thinning and skeletoni- zation [9]. A. Rosenfeld wrote the following: “The discrete grid (of pixels or voxels) used in digital topology can be regarded as a ‘digitization’ of (two or three-dimensional) Euclidean space; from this viewpoint, it is of interest to study conditions under which this digitization process preserves topological (or other geometric) properties” [3]. In this spirit, we obtain in this paper several properties of continuous maps on digital simple closed curves, inspired by analogs for Euclidean simple closed curves. In particular, we show that digital simple closed curves of more than 4 points are not contractible, and we obtain several results for continuous maps and antipodal points on digital simple closed curves. 2. Preliminaries Let Z be the set of integers. Then dZ is the set of lattice points in d-dimensional Euclidean space. Let dZX and let  be some adjacency relation for the members of X. Then the pair ),(X is said to be a (binary) digital image. A variety of adjacency relations are used in the study of digital images. Well known adjacencies include the following. For a positive integer l with 1ld and two distinct points 12 12=(,,, ), =(,,,),dddppppqqqqZp and q are lc-adjacent [10] if • there are at most l indices i such that 1|=| ii qp, and • for all other indices j such that 1||jjqp , jj qp =. See Figure 1. L. BOXER Copyright © 2010 SciRes. AM 378 Figure 1. In 2Z, each of the points 1= (1,0)p and 2= (0,1)p is both 1c-adjacent and 2c-adjacent to 0= (0,0)p, since both 1p and 2p differ from 0p by 1 in exactly one coordinate and coincide in the other coordinate; 3=(-1,1)p is 2c-adjacent, but not 1c-adjacent, to 0p, since 0p and 3p differ by 1 in both coordinates. The notation lc is sometimes also understood as the number of points dZq  that are lc-adjacent to a given point dZp . Thus, in Z we have 2=1c; in 2Z we have 4=1c and 8=2c; in 3Z we have 18,=6,= 21 cc and 26=3c. More general adjacency relations are studied in [11]. Let  be an adjacency relation defined on dZ. A - neighbor of a lattice point p is -adjacent to p. A digital image dZX is -connected [11] if and only if for every pair of different points Xyx ,, there is a set },,,{ 10 rxxx  of points of a digital image X such that rxyxx =,= 0 and ix and 1ix are -neighbors where 1},{0,1,  ri . Let Zba , with ba<. A digital interval [7] is a set of the form }.|{=],[ bzaZzba Z Let 0dZX and 1dZY be digital images with 0-adjacency and 1-adjacency respectively. A function YXf : is said to be ),( 10-continuous [2,8], if for every 0-connected subset U of X, )(Uf is a 1-connected subset of Y. We say that such a function is digitally continuous. Proposition 2.1 [2,8] Let 0dZX and 1dZY be digital images with 0-adjacency and 1-adjacency respectively. Then the function YXf : is ),(10- continuous if and only if for every 0-adjacent points },{10 xx of X, either )(=)( 10 xfxf or )(0xf and )( 1xf are 1-adjacent in Y. This characterization of continuity is what is called an immersion, gradually varied operator, or gradually varied mapping in [12,13]. Given digital images ),( iiX, {0,1}i, suppose there is a ),( 10-continuous bijection 10:XXf such that 011:XXf  is ),( 01-continuous. We say 0X and 1X are ),(10-isomorphic [14] (this was called ),( 10-homeomorphic in [7]) and f is a ),( 10-isomorphism (respectively, a ),( 10-homeo- morphism). By a digital -path from x to y in a digital image X, we mean a )(2,-continuous function Xmf Z][0,: such that xf =(0) and ymf=)( . We say m is the length of this path. A simple closed -curve of 4m points (for some adjacencies, the minimal value of m may be greater than 4; see below) in a digital image X is a sequence {(0), (1),,ff (1)}fm of images of the -path :[0, 1]Zfm X such that )(if and )( jf are -adjacent if and only if mij mod1)(=. If 10=}{=miixS where )(= ifx i for all Zmi 1][0,, we say the points of S are circularly ordered. Digital simple closed curves are often examples of digital images dZX for which it is desirable to consider XZ d\ as a digital image with some adja- cency (not necessarily the same adjacency as used by X). For example, by analogy with Euclidean topology, it is desirable that a digital simple closed curve 2ZX satisfy the “Jordan curve property” of separating 2Z into two connected components (one “inside” and the other “outside” X). An example that fails to satisfy this property [10] if we allow 4|=| X is given by ),(1cX , where (0,1)}(1,1),(1,0),{(0,0), = X is circularly ordered, and ),\(22cXZ is 2c-connected. However, this anomaly is essentially due to the “small- ness” of X as a simple closed curve; it is known [1,15,16] that for 8=1m and 4=2m, if 2ZX is a digital simple closed ic-curve such that imX|| , then XZ \2 has exactly 2 ic3-connected components, {1,2}i. Thus, it is customary to require that a digital simple closed curve 2ZX satisfy 8|| X when 1c- adjacency is used; 4|| X when 2c-adjacency is used. Let 0dZX and 1dZY be digital images with 0-adjacency and 1-adjacency respectively. Two ),( 10-continuous functions YXgf :, are said to be digitally ),(10-homotopic in Y [8] if there is a positive integer m and a function YmXH Z ][0,: such that • for all Xx, )(=,0)( xfxH and )(=),( xgmxH; • for all Xx, the induced function:[0, ]xZHm Y defined by ,][0,),(=)( Zx mtallfortxHtH is )(2, 1-continuous; and • for all Zmt ][0,, the induced function YXHt: defined by ,),(=)( XxallfortxHxHt is ),( 10-continuous. We say that the function H is a digital ),( 10- homotopy between f and g. L. BOXER Copyright © 2010 SciRes. AM 379Additional terminology associated with homotopic maps includes the following. • If g is a constant map, we say f is ),( 10- nullhomotopic [8]; if, further, ),(=),( 10YX and Xf1= (the identity map on X), we say X is 0- contractible [7,17]. • If )(=),( 00 xftxH for some Xx 0 and all Zmt ][0,, we say H is a ),( 10-pointed homotopy [18]. • If YmmHZZ  ][0,][0,: 10 is a ),( 1c-homotopy between ),( 1c-continuous functions Ymgf Z][0,:, 0 such that for all Zmt ][0, 1, (0)=(0)=)(0, gftH and )(=)(=),( 000 mgmftmH , we say H holds the endpoints fixed [18]. Proposition 2.2 [8] Suppose YXff :, 10 are ),(-continuous and ),(-homotopic. Suppose ZYgg:, 10 are ),(-continuous and ),(-ho- motopic. Then 00 fg  and 11 fg  are (,)-homo- topic in Z. 3. Homotopy Properties of Digital Simple Closed Curves A classical theorem of Euclidean topology, due to L.E.J. Brouwer, states that a d-dimensional sphere dS is not contractible [19]. Theorem 3.3, below, is a digital analog, for 1=d. We also present some related results in this section. Proposition 3.1 Let aS be a digital simple closed a-curve, {0,1}a. Let 10:SSf  be a ),( 10- continuous function. If ||=|| 10 SS , then the following are equivalent. a) f is one-to-one. b) f is onto. c) f is a ),( 10-isomorphism. Proof: Since ||=|| 10 SS , the equivalence of a) and b) follows from the fact that 0S is a finite set. That c) implies both a) and b) follows from the definition of isomorphism. Therefore, we can complete the proof by showing that b) implies c). Let 10=,}{=niiaa xS, where the points of aS are circu- larly ordered, {0,1}a. Let 11, Sx u and let )(= 1,10, uv xfx . Then the 1-neighbors of ux1, in 1S are nuxmod1)(1,  and nuxmod1)(1, , and the 0-neighbors of vx0, in 0S are nvxmod1)(0,  and nvxmod1)(0,. Since f is a continuous bijection, our choice of vx0, implies 0,(1) mod0,(1) mod1,(1) mod1,(1) mod{,} = {,}.vnvn ununfx xx x  Thus, 11,(1) mod1,(1) mod0,(1) mod0,(1) mod{,} = {,}.unun vnvnfx xx x  Since u was taken as an arbitrary index, 1f is ),( 01-continuous, so f is a ),( 10-isomorphism. Theorem 3.2 Let S be a simple closed -curve and let SmSH Z][0,: be a ),(-homotopy between an isomorphism 0H and =,mHfwhere SSf)( . Then 4|=| S. Proof: Let 10=}{=niixS , where the points of S are circularly ordered. There exists Zmw ][1, such that }.)(|][0,{min = SSHmtw tZ  Without loss of generality, )(1SHx w. Then the induced function 1wH is a bijection, so there exists Sxu such that 1=1),( xwxH u. By Proposition 3.1, },{=}),({ 20mod1)(mod1)(1 xxxxH nunuw  , and the continuity property of homotopy implies },{),( 20xxwxH u. Without loss of generality, (1)mod 0,1=unHxwx (1) and 2,=.uHxw x (2) Suppose 4>n. Equation (2) implies },,{),( 321mod1)( xxxwxH nu, but this is impossible, for the following reasons. • 1mod1)(),( xwxHnu, by choice of 1x. • },{),( 32mod1)(xxwxH nu, from Equation (1), because 4>n implies neither 2x nor 3x is -adjacent to 0x. The contradiction arose from the assumption that 4>n. Therefore, we must have 4n. Since a digital simple closed curve is assumed to have at least 4 points, we must have 4=n. In [8], an example is given of a simple closed 2c- curve 2ZS  such that SZ \2 has 2 1c-connected components, with 4|=| S, such that S is 2c-contrac- tible (see Figure 2). By contrast, we have the following. Theorem 3.3 Let ),(S be a simple closed -curve such that 4|>| S. Then S is not -contractible. Proof: It follows from Theorem 3.2 that if 4|>| S, then there cannot be a ),(-homotopy between S1 and a constant map in S. It is natural to ask whether we can obtain an analog of Theorem 3.3 for higher dimensions. In order to do so, we must decide what is an appropriate digital model for the k-dimensional Euclidean sphere kS. The literature contains the following. • Let kX2 be the set of all points kZp  such that p is a 1c-neighbor of the origin. Then ([10], Proposition 4.1) kX2 is kc-contractible. Notice that this example generalizes the contractibility of a 4-point digital simple closed curve [8] (see Figure 2 for the planar version); the contractibility seems due to the smallness of the image, rather than its form. • Let kIBd denote the boundary of a digital k-cube, i.e., for some integer 2>n, 1 = {(,,)[0,1]|{1,,},{0, 1}}.kkkZiBd Ixxnfor some ikxn  L. BOXER Copyright © 2010 SciRes. AM 380 Figure 2. 2c-contraction of 4X, a digital simple closed 2c-curve, via 44×[0,2]ZH:XX. (a) shows 4X. Points are labeled by indices. We have (,0)=Hx x for all 4xX. Arrows show the “motion” of 23,xx at the next step. (b) shows the results of the first step of the contraction: 030(,1) =,1) =Hx H(x x; 121( ,1)( ,1)Hx =Hx =x. The arrow shows the motion at the next step of the contraction. (c) shows the results of the final step of the contraction: 0(2)iHx, =x for all {0,1, 2, 3}i. See Figure 3 for the planar version. Then ([7], Coro- llary 5.9) for >2n, kBdI is not 1c-contractible. The next result may be interpreted as stating that for 4>n, a map homotopic to S1 must be a “rotation” of the points of S. Theorem 3.4 Let S be a simple closed -curve such that 4>|=| nS . Let SSf : be a ),(- continuous function such that f is ),(-homotopic to S1. Then, for some integer j, we have .1][0, = )(mod)(Znjii niallforxxf  Proof: Let SmSH Z][0,: be a ),(-homo- topy from S1 to f. For Zmt ][0,, let SSHt: be the induced map. The assertion follows from the following. Claim 1: For each Zmt ][0,, there is an integer j such that njiit xxH mod)(=)(  for all Zni 1][0,. To prove Claim 1, we argue by mathematical induc- tion on t. For 0=t, we can clearly take 0=j. Now, suppose the claim is valid for Zut ][0, such that mu <0 . Then, in particular, there is an integer j such that njiiu xxH mod)(=)( for all Zni 1][0,. The continuity properties of homotopy imply 10()uHx ( 1)mod( 1)mod{,, }jnj jnxxx. Without loss of generality, juxxH =)(01. This is the initial case of the following: Claim 2: njkku xxH mod)(1 =)( for all Znk 1][0,. Suppose the equation of Claim 2 is true for all Zvk ][0,, for some Znv 2][0,. In particular, .=)(mod)(1 njvvu xxH  (3) By the continuity properties of homotopy, )( 11 vu xH is adjacent to or coincides with njvvuxxH mod)(1 =)(  and with 1(1)mod()= .uvv jnHxx Thus, 11 ()mod(){ ,uv vjnHxx (1)mod}vj nx . Since 1uH must also be an isomorphism by Theorem 3.2, from Equation (3), 11()=uvHx (1)modvj nx . This completes the induction proof for the Claim 2, which, in turn, completes the induction proof for the Claim 1. Thus, the assertion is established. 4. Antipodal Maps A classical theorem of Euclidean topology, due to K. Borsuk, states that a continuous antipodal map:dfS dS from the d-dimensional unit sphere to itself is not homotopic to a constant map [19]. In this section, we obtain a digital analog, Theorem 4.16, for 1=d. We say a set dZX is symmetric with respect to the origin if X satisfies the property that .XxifonlyandifXx  Suppose we have 0dXZ, 1dYZ, and X is sym- metric with respect to the origin. A function :fXY is called antipodal-preserving or an Figure 3. 2BdI , the “boundary” of a digital square. L. BOXER Copyright © 2010 SciRes. AM 381 Figure 4. A digital simple closed 1c-curve 7=0={ }iiSx and a 11(,)cc-continuous map :fSS such that f is 11(,)cc -homotopic to 1S. According to Theorem 3.4, such a map f must “rotate” the members of S. In (a), points of S are labeled by indices. In (b), each yS is labeled by the index of the point ixS such that ()=ifxy. Here, we have (2)mod8()=iifx x for all i. antipodal map if )(=)( xfxf  for all Xx  [19]. In this section, we study properties of continuous antipodal maps between digital simple closed curves. Lemma 4.1 Let 10,pp be lc-adjacent points in dZ, dl 1. Then 0p and 1p are not antipodal. Proof: The hypothesis implies that there is an index i such that 0p and 1p differ by 1 in the thi coordinate: 1|=| 1,0, iipp. If 0p and 1p are antipodal, this would imply 1/2,1/2}{=},{ 1,0,ii pp , which is impossible. Lemma 4.2 Let 10,pp be lc-adjacent points in dZ, dl 1. Then 0p and 1p are lc-adjacent. Proof: Elementary, and left to the reader. Lemma 4.3 Let 10=}{=niixS be a digital simple closed lc-curve in dZ such that the points of S are cir- cularly ordered. If S is symmetric with respect to the origin, then the origin is not a member of S. Proof: Suppose the origin is a member of S. Without loss of generality, 0x is the origin, and therefore is its own antipode. By Lemma 4.2, 1x and 1nx are antipodes; by Lemma 4.1, these points are not lc-adjacent. This establishes the base case of an induction argument: 11=10=1 }{}{=jjnii xxS  is a connected subset of S, such that 1x and 1nx are non-adjacent antipodes; hence, 1S is ),(1ccl-isomorphic to a digital interval. Now, suppose for some integer k, 1)/2(<1 nk , kjjnkiik xxS 1=0= }{}{=  is ),(1ccl-isomorphic to a digital interval, with endpoints kx and knx, such that mx and mnx are non-adjacent antipodes for all },{1, km. Then, by Lemma 4.2, 1kx and 1knx are antipodes, and by Lemma 4.1, these points are not lc-adjacent. Thus, 11=10=1 }{}{= kjjnkiik xxS is ),(1ccl- isomorphic to a digital interval, with endpoints 1x and 1nx. This completes an induction argument from which we conclude that 11)/2(=1)/2(0= }{}{ =  nnniinii xxS is ),(1ccl-isomorphic to a digital arc, with the endpoints of S being  1)/2(nx and  1)/2(nnx, such that  1)/2(1 ni implies ix and inx are non-adjacent antipodes. • If n is odd, then 1n is even, so SS =. This is a contradiction, since S is not a simple closed lc- curve. • If n is even, then }{\= /2nxSS. Since S is symmetric with respect to the origin, we must have that /2nx is the origin. But since S is a simple closed lc-curve in which 0x is the origin, this is a contra- diction. Whether n is even or odd, the assumption that the origin belongs to S yields a contradiction. Hence, the origin is not a member of S. Lemma 4.4 Let 10=}{=niixS be a digital simple closed lc-curve in dZ such that the points of S are circularly ordered. If S is symmetric with respect to the origin, then n is even. Proof: By Lemma 4.3, the origin is not a member of S, so every member of S is distinct from its antipode. Therefore, n must be even. Lemma 4.5 Let 10=}{=niixS be a digital simple closed lc-curve in dZ such that the points of S are cir- cularly ordered. If S is symmetric with respect to the origin, then for all i we have nniixx mod/2)(=. Proof: Suppose there is a simple closed lc-curve 10=}{=niixS that is symmetric with respect to the origin such that the points of S are circularly ordered, such that there exist indices vu, such that ux and vx are antipodes and nnuv mod/2)(. Without loss of generality, we can assume 0=u, /2nv . Then, from Lemma 4.2, 1x is antipodal to either 1vx or nvxmod1)( . Without loss of generality, 1x and 1vx are antipodal. If 1=1v, this is a contradiction of Lemma 4.3; or, if 2=1v, this is a contradiction of Lemma 4.1; otherwise, we inductively repeat the argument above with the antipodal (by Lemma 4.2) pair ),( 22vxx , etc., until similarly we obtain a contradiction of Lemma 4.3 or of Lemma 4.1. The assertion follows. We have the following. Theorem 4.6 Let idiZS be simple closed i-curves, {0,1}i, each symmetric with respect to the origin. Let 10:SSf  be a ),( 10-continuous antipodal map. Then f is onto. Proof: Let 10=1 }{=niixS , where the points of 1S are circularly ordered. Without loss of generality, there exists 0Sp such that 0=)( xpf . Since f is anti- L. BOXER Copyright © 2010 SciRes. AM 382 podal, it follows from Lemma 4.5 that /2=)( nxpf. Since f is continuous and 0S is 0-connected, it follows that one of the 1-paths in 1S from 0x to /2nx is contained in )( 0Sf . Without loss of generality, )(}{ 0/20= Sfx njj. For njn <| Y, then f is not ),( 10-nullhomotopic. Proof: Suppose there is such a function f that is ),( 10-nullhomotopic. From Lemma 4.9, f is point- ed homotopic to )( 0xf . Let Xnb ZX ][0,: be defined by Xntxtb mod=)( . Let YZp : be the ),( 1Yc-covering map defined by Ynzyzp mod=)( (see Example 4.12). By Proposition 2.2, bf  and bxf)( 0 are ),( 1Yc-homotopic paths in Y. From Theorem 4.14, these functions have unique Figure 5. A simple closed 1c-curve C and a covering by the digital line Z. Members of C are labeled by their respective indices. A point zZ is labeled by the index of point of C to which the covering map sends z. L. BOXER Copyright © 2010 SciRes. AM 384 Figure 6. Example of lifting. 5=0={ }iiBb is a simple closed 2c-curve whose members are labeled by their indices. =EZ has its points z labeled above by their coordinates and labeled below by the index i such that ()=ipzb (note p is given by the formula mod 6()=zpz b). 11=0={ }mmXx is a simple closed 2c-curve that has points labeled by a pair ,mn such that m is the index of the point, ()=mnfxb (thus, f is defined by mod 6()=mmfx b), and ()=mFxm. Since p is a covering map (by Example 4.12) and =pF f, F is a lifting of f with respect to p. liftings with respect to p to paths 0F and 1F, respectively, in Z, each starting at (0)))((0 1bfp. Since 5|>| Y, YxfN Y),2)(( 0, so p is a radius 2 local isomorphism. From Theorem 4.15, 0F and 1F must end at the same point. Indeed, this point must be 0, for the uniqueness of 1F implies 1F must be the cons- tant map 0. Since f is an antipodal map, we must have .1]/2[0,/2 = |)(/2)(|00 ZXYXntallforntFntF (4) Since 0=(0)0F, /2}/2,{/2)(0YYXnnnF. Without loss of generality, /2.=/2)(0YX nnF (5) Since F is continuous and 5>Yn, a simple induc- tion argument based on Equations (4) and (5) shows that )(>/2)( 00tFntFX for all ZXnt1]/2[0, . But since 0=)(0XnF , this implies with Equation (4) that /2=/2)(0YX nnF , which contradicts Equation (5). The assertion follows from the contradiction. 5. Antipodes Mapped Together A classical result of topology is that if f is a conti- nuous map from the d-dimensional unit sphere dS to Euclidean d-space dR, then there is a pair of antipodes dSxx , such that )(=)( xfxf  [17]. For 1=d, the following is a digital analog. Theorem 5.1 Let S be a digital simple closed lc-curve with 10=}{=niixS , where the points of S are circularly ordered. Suppose S is symmetric with respect to the origin. Suppose ZSf: is a ),( 1ccl- continuous function. Then there is a pair of antipodes Sxx, such that 1|)()(|xfxf . Proof: By Lemma 4.4, n is even. Consider the function ZngZ1][0,: defined by()=( )igif x (/2)mod()in nfx. By Lemma 4.5, we are done if, for some i, 0=)(ig . Therefore, we assume for all i that 0.)(ig (6) Clearly, for all i, ).(=]mod/2)[( ignnig  (7) The continuity of f implies that 2.|1)()(| igig (8) It follows from Equation (7) that g takes both posi- tive and negative values, so from inequality (6), there is an index j such that )( jg and 1)( jg have oppo- site sign; without loss of generality, 0>)(jg and 0<1)(jg . From inequality (8), it follows that L. BOXER Copyright © 2010 SciRes. AM 385 Figure 7. S and :fSZ. Each number in the grid labels a point of S, showing the image of the grid point under f. Note for 0= (1,2)s, 00|()( )|=1fsf s , but there is no sS for which ()= ()fsfs. 1=)( jg , and the assertion follows. Theorem 5.1 parallels, in a sense, a result of [2]: In the Euclidean line, a continuous function mapping an interval to itself has a fixed point; in the digital world, a ),( 11cc - continuous function ZZbabaf ],[],[:  has a “near- fixed” point, i.e. , a point x such that 1|)(|xfx . That we cannot, in general, conclude the existence of antipodes mapped to the same point in Theorem 5.1, is illustrated in the following example (note the simple closed curve has more than 4 points). Let=S {(,)|||||=3} xy xy. Then S is a simple closed 2c- curve in 2Z. The function ZSf : given by 3,=(0,3)=1,2)(=2,1)(=3,0)( ffff  2,=(1,2)=1)2,( ff  1,=(2,1)=2)1,( ff  0,=(3,0)=1)(2,=2)(1,=3)(0, ffff  is a ),( 12cc -continuous function such that ()( )fxfx 0 for each Sx. See Figure 7. 6. Further Remarks In this paper, we have obtained several analogs of classical theorems of Euclidean topology concerning maps on digital simple closed curves. 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