Applied Mathematics, 2010, 1, 377-386
doi:10.4236/am.2010.15050 Published Online November 2010 (http://www.SciRP.org/journal/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
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 l
c-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 d
Z
is the set of
lattice points in d-dimensional Euclidean space. Let
d
Z
X
 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
=(,,, ), =(,,,),
d
dd
ppppqqqqZ p
and q are l
c-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 378
Figure 1. In 2
Z
, each of the points 1= (1,0)p and
2= (0,1)p is both 1
c-adjacent and 2 c-adjacent to
0= (0,0)p, since both 1
p and 2 p differ from 0 p by 1 in
exactly one coordinate and coincide in the other coordinate;
3=(-1,1)p is 2
c-adjacent, but not 1 c-adjacent, to 0 p,
since 0
p and 3 p differ by 1 in both coordinates.
The notation l c is sometimes also understood as the
number of points d Zq  that are l c-adjacent to a
given point d Zp . Thus, in
Z
we have 2=
1
c; in
2
Z
we have 4=
1
c and 8=
2
c; in 3
Z
we have
18,=6,= 21 cc and 26=
3
c.
More general adjacency relations are studied in [11].
Let

be an adjacency relation defined on d
Z
. A

-
neighbor of a lattice point p is

-adjacent to p. A
digital image d
Z
X
 is

-connected [11] if and only
if for every pair of different points Xyx ,, there is a
set },,,{ 10 r xxx  of points of a digital image
X
such
that r xyxx =,= 0 and i x and 1i x are

-neighbors
where 1},{0,1,  ri .
Let Zba , with ba<. A digital interval [7] is a
set of the form
}.|{=],[ bzaZzba Z
Let 0
d
Z
X
 and 1
d
Z
Y
 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 0
d
Z
X
 and 1
d
Z
Y
 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 )(0 xf and
)( 1 xf 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 ),( ii X

, {0,1}i, suppose
there is a ),( 10


-continuous bijection 10 :XXf
such that 01
1:XXf   is ),( 01


-continuous. We
say 0 X and 1 X 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]
Z
fm

X
 such that )(if and )( jf are

-adjacent if
and only if mij mod1)(=

. If 1
0=
}{=m
ii
xS where
)(= ifx i for all Z mi 1][0,


, we say the points of S
are circularly ordered.
Digital simple closed curves are often examples of
digital images d
Z
X
 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 2
Z
X

satisfy the “Jordan curve property” of separating 2
Z
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 ),(1 cX ,
where
(0,1)}(1,1),(1,0),{(0,0), = X
is circularly ordered, and ),\(2
2cXZ is 2
c-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=
1
m and 4=
2
m, if 2
Z
X
 is a
digital simple closed i c-curve such that i mX|| , then
XZ \
2 has exactly 2 i
c3-connected components,
{1,2}

i. Thus, it is customary to require that a digital
simple closed curve 2
Z
X
 satisfy 8|| X when 1 c-
adjacency is used; 4|| X when 2 c-adjacency is used.
Let 0
d
Z
X
 and 1
d
Z
Y
 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, ]
x
Z
H
m
Y defined by
,][0,),(=)( Zx mtallfortxHtH

is )(2, 1

-continuous; and
• for all Z mt ][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 379
Additional 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
X
f1= (the identity map on
X
), we say
X
is 0

-
contractible [7,17].
• If )(=),( 00 xftxH for some Xx 
0 and all
Z
mt ][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 Z mt ][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 d
S 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 a S 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 0 S 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 1
0=,}{=n iiaa xS, where the points of a
S are circu-
larly ordered, {0,1}a. Let 11, Sx u and let
)(= 1,
1
0, uv xfx . Then the 1

-neighbors of u x1, in 1 S
are nu xmod1)(1,  and nu xmod1)(1, , and the 0

-neighbors of
v
x0, in 0 S are nv xmod1)(0,  and nv xmod1)(0,. Since f is a
continuous bijection, our choice of v x0, implies

0,(1) mod0,(1) mod1,(1) mod1,(1) mod
{,} = {,}.
vnvn unun
fx xx x
 
Thus,

1
1,(1) mod1,(1) mod0,(1) mod0,(1) mod
{,} = {,}.
unun vnvn
fx 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 0 H and =,
m
H
fwhere
SSf

)( . Then 4|=| S.
Proof: Let 1
0=
}{=n
ii
xS , where the points of S are
circularly ordered.
There exists Z mw ][1,

such that
}.)(|][0,{min = SSHmtw tZ 

Without loss of generality, )(
1SHx w

. Then the
induced function 1w
H 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=
un
H
xwx
 (1)
and


2
,=.
u
H
xw x (2)
Suppose 4>n. Equation (2) implies
},,{),( 321mod1)( xxxwxH nu

, but this is impossible, for
the following reasons.
• 1mod1)(),( xwxHnu

, by choice of 1
x.
• },{),( 32mod1)(xxwxH nu


, from Equation (1), because
4>n implies neither 2 x nor 3 x is

-adjacent to 0 x.
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 2 c-
curve 2
ZS  such that SZ \
2 has 2 1
c-connected
components, with 4|=| S, such that S is 2
c-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 S 1
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 k S. The literature
contains the following.
• Let k X2 be the set of all points k Zp  such that
p is a 1 c-neighbor of the origin.
Then ([10], Proposition 4.1) k X2 is k c-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 k
IBd denote the boundary of a digital k-cube,
i.e., for some integer 2>n,
1
= {(,,)[0,1]|{1,,},
{0, 1}}.
k
kkZ
i
Bd Ixxnfor some ik
xn
 



L. BOXER 380
Figure 2. 2 c-contraction of 4
X
, a digital simple closed 2 c-curve, via 
44
×[0,2]Z H:XX. (a) shows 4
X
. Points are
labeled by indices. We have (,0)=Hx x for all 4
x
X

. Arrows show the “motion” of 23 ,
x
x 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)
i
Hx, =x for all {0,1, 2, 3}i.
See Figure 3 for the planar version. Then ([7], Coro-
llary 5.9) for >2n, k
B
dI is not 1 c-contractible.
The next result may be interpreted as stating that for
4>n, a map homotopic to S 1 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 S
1. Then, for some integer j, we have
.1][0, = )(mod)(Znjii niallforxxf 

Proof: Let SmSH Z][0,: be a ),(


-homo-
topy from S
1 to f. For Z mt ][0,

, let SSHt:
be the induced map. The assertion follows from the
following.
Claim 1: For each Z
mt ][0,, there is an integer j
such that njiit xxH mod)(
=)(  for all Z ni 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 Z
ut ][0, such that
mu <0 . Then, in particular, there is an integer j
such that njiiu xxH mod)(
=)( for all Z ni 1][0,. The
continuity properties of homotopy imply 10
()
u
Hx


( 1)mod( 1)mod
{,, }
j
nj jn
xxx

.
Without loss of generality, juxxH =)(01. This is the
initial case of the following:
Claim 2: njkku xxH mod)(1 =)( for all Z
nk 1][0,

.
Suppose the equation of Claim 2 is true for all
Z
vk ][0,, for some Z nv 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 jn
Hxx
 Thus, 11 ()mod
(){ ,
uv vjn
Hxx


(1)mod
}
vj n
x . Since 1u H must also be an isomorphism
by Theorem 3.2, from Equation (3), 11
()=
uv
Hx

(1)modvj n
x . 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:d
f
S
d
S 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 d
Z
X
 is symmetric with respect to
the origin if
X
satisfies the property that
.XxifonlyandifXx 

Suppose we have 0
d
X
Z, 1
d
YZ, and
X
is sym-
metric with respect to the origin. A function :
f
XY
is called antipodal-preserving or an
Figure 3. 2
B
dI , the “boundary” of a digital square.

L. BOXER 381
Figure 4. A digital simple closed 1 c-curve 7
=0
={ }
ii
Sx and a
11
(,)cc-continuous map :
f
SS 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 i
x
S such that ()=
i
f
xy. Here, we
have (2)mod8 ()=
ii
fx 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 l c-adjacent points in d
Z
,
dl 1. Then 0 p and 1 p are not antipodal.
Proof: The hypothesis implies that there is an index i
such that 0
p and 1 p differ by 1 in the th
i coordinate:
1|=| 1,0, iipp. If 0 p and 1 p are antipodal, this would
imply 1/2,1/2}{=},{ 1,0,
ii pp , which is impossible.
Lemma 4.2 Let 10,pp be l c-adjacent points in d
Z
,
dl 1. Then 0 p and 1 p are l c-adjacent.
Proof: Elementary, and left to the reader.
Lemma 4.3 Let 1
0=
}{=n
ii
xS be a digital simple closed
l
c-curve in d
Z
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, 0
x is the origin, and therefore is its
own antipode.
By Lemma 4.2, 1
x and 1n x are antipodes; by
Lemma 4.1, these points are not l
c-adjacent. This
establishes the base case of an induction argument:
1
1=
1
0=1 }{}{=jjnii xxS 
 is a connected subset of S, such
that 1 x and 1n x are non-adjacent antipodes; hence,
1
S is ),(1 ccl-isomorphic to a digital interval. Now,
suppose for some integer k, 1)/2(<1 nk ,
k
jjn
k
iik xxS 1=0= }{}{= 
 is ),(1 ccl-isomorphic to a digital
interval, with endpoints k x and kn x, such that m x
and mn x are non-adjacent antipodes for all
},{1, km. Then, by Lemma 4.2, 1k x and 1kn x
are antipodes, and by Lemma 4.1, these points are not
l
c-adjacent. Thus, 1
1=
1
0=1 }{}{=  

k jjn
k
iik xxS is ),(1
ccl-
isomorphic to a digital interval, with endpoints 1
x and
1n
x.
This completes an induction argument from which we
conclude that
1
1)/2(=
1)/2(
0= }{}{ = 

 
n
nni
i
n
ii xxS
is ),(1 ccl-isomorphic to a digital arc, with the endpoints
of S

being  1)/2(n x and  1)/2(nn x, such that
 1)/2(1 ni implies i x and in x 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 l c-
curve.
• If n is even, then }{\= /2n xSS. Since S is
symmetric with respect to the origin, we must have that
/2n
x is the origin. But since S is a simple closed
l
c-curve in which 0 x 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 1
0=
}{=n
ii
xS be a digital simple closed
l
c-curve in d
Z
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 1
0=
}{=n
ii
xS be a digital simple closed
l
c-curve in d
Z
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 l c-curve
1
0=
}{=n
ii
xS 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 u x and v x are
antipodes and nnuv mod/2)(


. Without loss of
generality, we can assume 0=u, /2nv .
Then, from Lemma 4.2, 1 x is antipodal to either 1v x
or nv
xmod1)( . Without loss of generality, 1 x and 1v x
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v xx , 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 i
d
iZS 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 1
0=1 }{=n ii
xS , where the points of 1 S are
circularly ordered. Without loss of generality, there
exists 0
Sp

such that 0 =)( xpf . Since f is anti-

L. BOXER 382
podal, it follows from Lemma 4.5 that /2 =)( n xpf.
Since f is continuous and 0 S is 0

-connected, it
follows that one of the 1

-paths in 1 S from 0 x to
/2n
x is contained in )( 0 Sf . Without loss of generality,
)(}{ 0
/2
0= Sfx n jj. For njn <</2 , there exists 0
Sq

such that /2 =)( nj xqf , and from Lemma 4.5,
/2
0
= = ()
= ()()().
jjn
xx fq
s
incefis antipodalfqfS



Thus, 10 =)( SSf .
Corollary 4.7 Let i
d
iZS be simple closed i

-
curves, {0,1}i, each symmetric with respect to the
origin. Let 10
:SSf  be a ),( 10


-continuous anti-
podal map. If ||=|| 10 SS , then f is a ),(


-iso-
morphism.
Proof: By Theorem 4.6, f is onto. The assertion
follows from Proposition 3.1.
Proposition 4.8 Let 1
0=
}{=
X
n
ii
xX be a simple closed
X

-curve with circularly ordered points. Let=Y
1
=0
{}
nY
ii
y be a simple closed Y

-curve with circularly
ordered points. Suppose YmXH Z ][0,: is a
),( YX


-homotopy between the ),( YX


-continuous
maps m
HH ,
0 such that )(=)(000 xHxHm. Then there
is a ),(YX


-homotopy :[0,]
Z
GX m Y between
0
H and m H such that0 (,)=Gxt 00 ()
H
x for all
Z
mt ][0,
Proof: Without loss of generality, 000 =)( yxH . Let
ZZ mma ][0,][0,:  be defined by ita =)( if
i
ytxH =),(0.
Let YmXG Z][0,: be defined by
.=),(, = ),(mod)]([ ji
Y
ntaji ytxHifytxG 
Roughly, we may think of ),( txGi as rotating
),(txH i counter to the rotation of ),(0txH , so that the
image under G of 0 x at time
t
is constant with
respect to
t
. We show below that G is a homotopy. In
particular, 0=)(=(0) maa , so 00 =HG and
mm HG =; and, for all Z
mt ][0,,
0[()()]mod0
(,) = = .
at atn
Y
Gx tyy

For each Xxi and Z mt ][0,

, if jit yxH =)(
then we have the following.
• By the continuity of t
H, we have
( 1)mod(1)mod(1)mod(1)mod
({,}) {, ,}.
tin injnjjn
XXY Y
Hx xyyy
 

Therefore, [()]mod ()=
tijatn
Y
Gx y
 and
 


 

1mod1mod
1 modmod1 mod
,
,, .
tini n
XX
jatn jatn jatn
YY Y
Gxx
yyy

  
  

Therefore, t
G is ),( YX


-continuous.
• Let YmG Z
i
x][0,: be the induced function de-
fined by ),(=)(txGtG i
i
x. To simplify the following, let
10
(1)=( )=( ),
x
ii
i
GHxHx


1
(1)= ()=(),
x
mi mi
i
GmH xHx


1)(=)(=0=(0)=1)( 

mamaaa . Note that the con-
tinuity of 0
x
H implies that
{(1),(1)}{()1,( ),( )1}.atatatat at

 
Suppose jit yxH =)(, so [()]mod
()=
x
jat n
iY
Gty
.
If 1)(=1)(


tata , then
[(1)]mod [()1]mod
(1)==
x
jat njatn
iYY
Gt yy
 

is Y

-adjacent
to ()
xi
Gt
.
If )(=1)( tata

, then )(=1)( tGtG i
x
i
x.
If 1)(=1)(


tata , then
[(1)]mod [()1]mod
(1)==
x
jat njatn
iYY
Gt yy
 

is Y

-adjacent
to )(tGi
x.
Similarly, 1)(

tGi
x is Y

-adjacent to, or equal to,
().
xi
GtTherefore, the induced function i
x
G is ),( 1Y c

-
continuous.
Therefore, G is a homotopy between 0 H and m H
such that )(=),( 000 xHtxG for all Z mt ][0,.
Let ),( X X

and ),(Y Y

be digital images and let
Yy

. We denote by :
y
XY (or, for short,
y
) the
constant map ()=yx y for all Xx .
Lemma 4.9 Let ),( X X

and ),( Y Y

be digital
simple closed curves and let YXf : be ),( YX


-
homotopic to a constant map. Then for any Xx

0,
))(,(),(:00 xfYxXf  is ),( YX


-pointed homotopic
to the constant map )( 0 xf .
Proof: Let YmXF Z

][0,: 0 be a ),(YX


-
homotopy between f and the constant map 0 y for
some Yy

0. Since Y is Y

-connected, there is a
path Ymp Z][0,:1 from 0 y to )( 0 xf . Then the
map YmmXH Z


][0,: 10 defined by





,)(
;0),(
=),(
1000
0
mmtmifmtp
mtiftxF
txH
is a homotopy between f and )(0
xf . The assertion
follows from Proposition 4.8.
Given a digital image ),(

X and a positive integer
n, for Xx

we define [20]
(, )= {}
{|
}.
Nxn x
yXthereisapath inX
f
romytoxoflength at mostn




The covering space and the lifting of maps are notions
borrowed from algebraic topology that have been
important in digital algebraic topology. We have the
following.
Definition 4.10 [20] Let ),( E E

and ),( B B

be
digital images. Let BEp : be a ),( BE


-conti-
nuous function. Suppose for each Bb  there exists
N


such that

L. BOXER 383
• for some N

and some index set
M
,
);(),(=)),(( 11 bpewitheNbNpii
E
Mi
B





• if Mji ,, ji , then
 =),(),(


 j
E
i
EeNeN ; and
• the restriction map ),(),(:| ),( 


bNeNp B
i
Ei
e
E
N
is a ),( BE


-isomorphism for all Mi .
Then the map p is a ),( BE


- covering map, and
the pair ),( pE is a ),( BE


- covering (or covering
space).
The following is a somewhat simpler characterization
of the digital covering than given in Definition 4.10.
Theorem 4.11 [14] Let ),( E E

and ),(B B

be
digital images. Let BEp : be a ),( BE


-
continuous function. Then the map p is a ),( BE


-
covering map if and only if for each Bb , there is an
index set
M
such that
• ,1)(=,1))((
1
i
E
Mi
BeNbNp
 
, with )( 1bpei 
;
• if Mji ,, ji , then

=,1)(,1)( j
E
i
EeNeN

;
and
• the restriction map ,1)(,1)(:|,1)( bNeNpB
i
Ei
e
E
N 


is a ),( BE


-isomorphism for all Mi .
The following is a minor generalization of an example
of a digital covering map given in [20].
Example 4.12 Let dm
ii ZcC 
1
0=
}{= be a circularly
ordered simple closed

-curve. Let CZp : be
defined by mzzp mod=)( for all
Z
z
. Then p is
a ),( 1

c-covering map (see Figure 5).
Definition 4.13 [20] For digital images ),( E E

,
),( B B

, and ),( X X

, let BEp : be a ),( BE


covering map, and let BXf: be ),( BX


-
continuous. A lifting of f with respect to p is a
),( EX


-continuous function EXF : such that
fFp =.
See Figure 6 for an illustration of Definition 4.13.
Theorem 4.14 [20] Let ),( E E

be a digital image
and Ee

0. Let ),(B
B

be a digital image and
Bb

0. Let BEp: be a ),( BE


-covering map
such that 00 =)( bep. Then a B

-path BmfZ][0,:
beginning at 0
b has a unique lifting with respect to p
to a path f
~ in E starting at 0
e.
For a positive integer n, a ),( BE


-covering map
BEp : is called a radius n local isomorphism
[21] if for each Bb

and 1(),epb

(,) |
N
en
E
p

(,)
B
Nbn

 is an isomorphism. It was observed in [14]
that every covering is a radius 1 local isomorphism, but
there are coverings that are not radius 2 local isomor-
phisms.
Theorem 4.15 [21] Let ),( E E

be a digital image
and Ee

0. Let ),(B
B

be a digital image and
Bb

0. Let BEp : be a ),(BE


-covering map
such that 00 =)( bep . Suppose p is a radius 2 local
isomorphism. For E

-paths Emgg Z][0,:, 10 that
start at 0 e, if there is a B

-homotopy in B from
0
gp  to 1 gp  that holds the endpoints fixed, then
)(=)( 10 mgmg , and there is a E

-homotopy in
E
from 0 g to 1 g that holds the endpoints fixed.
Theorem 4.16 Let 1
0=
}{=
X
n
ii
xX be a simple closed
X

-curve with points circularly ordered, that is
symmetric with respect to the origin. Let 1
0=
}{= 
Y
n
ii
yY be
a simple closed Y

-curve with points circularly ordered,
that is symmetric with respect to the origin. Let
YXf : be a ),( YX


-continuous antipodal map. If
5|>| 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 )( 0 xf .
Let Xnb ZX ][0,: be defined by X
nt
xtb mod =)( .
Let YZp : be the ),( 1Y c

-covering map defined
by Y
nz
yzp mod =)( (see Example 4.12). By Proposition
2.2, bf  and bxf)( 0 are ),( 1Y
c

-homotopic paths
in
Y
. From Theorem 4.14, these functions have unique
Figure 5. A simple closed 1 c-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 384
Figure 6. Example of lifting. 5
=0
={ }
ii
Bb is a simple closed 2 c-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 ()=i
p
zb (note
p
is given by
the formula mod 6 ()=z pz b). 11
=0
={ }
mm
Xx is a simple closed 2 c-curve that has points labeled by a pair ,mn such that m is
the index of the point, ()=
mn
f
xb (thus, f is defined by mod 6 ()=
mm
fx b), and ()=
m
F
xm . 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 0 F and 1 F,
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, 0 F and 1 F
must end at the same point. Indeed, this point must be 0,
for the uniqueness of 1 F implies 1 F 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)
0
F, /2}/2,{/2)(
0YYXnnnF

. Without
loss of generality,
/2.=/2)(
0YX nnF (5)
Since
F
is continuous and 5>
Y
n, a simple induc-
tion argument based on Equations (4) and (5) shows that
)(>/2)( 00tFntFX  for all ZX nt1]/2[0, 

. But since
0=)(
0X
nF , 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 d S to
Euclidean d-space d R, then there is a pair of antipodes
d
Sxx , such that )(=)( xfxf  [17]. For 1=d,
the following is a digital analog.
Theorem 5.1 Let S be a digital simple closed
l
c-curve with 1
0=
}{=n
ii
xS , where the points of S are
circularly ordered. Suppose S is symmetric with
respect to the origin. Suppose ZSf: is a ),( 1
ccl-
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()=( )
i
gif x

(/2)mod
()
in n
fx
. 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 385
Figure 7. S and :
f
SZ. 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
s
S for which ()= ()
f
sfs.
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 2 c-
curve in 2
Z
. 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 ()( )
f
xfx


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. We have shown
that digital simple closed curves of more than 4 points
are not contractible; that a continuous antipodal map
from a digital simple closed curve to itself is not
nullhomotopic; and that a continuous map from a digital
simple closed curve to the digital line must map a pair of
antipodes within 1 of each other. Except as indicated
concerning whether or not a digital model of a sphere is
contractible, it is not known at the current writing
whether these results extend to higher dimensional
digital models of Euclidean spheres.
We thank the anonymous referees for their helpful
suggestions.
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