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@ Appl. Gen. Topol. 18, no. 2 (2017), 401-427doi:10.4995/agt.2017.7798
c© AGT, UPV, 2017

Generalized normal product adjacency in

digital topology

Laurence Boxer

Department of Computer and Information Sciences, Niagara University, Niagara University, NY

14109, USA; and Department of Computer Science and Engineering, State University of New

York at Buffalo. (boxer@niagara.edu)

Communicated by S. Romaguera

Abstract

We study properties of Cartesian products of digital images for which

adjacencies based on the normal product adjacency are used. We show

that the use of such adjacencies lets us obtain many “product proper-

ties” for which the analogous statement is either unknown or invalid if,

instead, we were to use cu-adjacencies.

2010 MSC: 54H99.

Keywords: digital topology; digital image; continuous multivalued function;

shy map; retraction.

1. Introduction

We study adjacency relations based on the normal product adjacency for
Cartesian products of multiple digital images. Most of the literature of digital
topology focuses on images that use a cu-adjacency; however, the results of
this paper seem to indicate that for Cartesian products of digital images, the
natural adjacencies to use are based on the normal product adjacency of the
factor adjacencies, in the sense of preservation of many properties in Cartesian
products.

Received 27 June 2017 – Accepted 25 July 2017

http://dx.doi.org/10.4995/agt.2017.7798


2. Preliminaries

We use N, Z, and R to represent the sets of natural numbers, integers, and
real numbers, respectively.

Much of the material that appears in this section is quoted or paraphrased
from [15, 17], and other papers cited in this section.

We will assume familiarity with the topological theory of digital images. See,
e.g., [3] for many of the standard definitions. All digital images X are assumed
to carry their own adjacency relations (which may differ from one image to
another). When we wish to emphasize the particular adjacency relation we
write the image as (X, κ), where κ represents the adjacency relation.

2.1. Common adjacencies. Among the commonly used adjacencies are the
cu-adjacencies. Let x, y ∈ Z

n, x 6= y. Let u be an integer, 1 ≤ u ≤ n. We say
x and y are cu-adjacent if

• there are at most u indices i for which |xi − yi| = 1, and
• for all indices j such that |xj − yj| 6= 1 we have xj = yj.

We often label a cu-adjacency by the number of points adjacent to a given
point in Zn using this adjacency. E.g.,

• In Z1, c1-adjacency is 2-adjacency.
• In Z2, c1-adjacency is 4-adjacency and c2-adjacency is 8-adjacency.
• In Z3, c1-adjacency is 6-adjacency, c2-adjacency is 18-adjacency, and
c3-adjacency is 26-adjacency.

Given digital images or graphs (X, κ) and (Y, λ), the normal product ad-
jacency NP(κ, λ) (also called the strong adjacency [37] and denoted κ∗(κ, λ)
in [13]) generated by κ and λ on the Cartesian product X × Y is defined as
follows.

Definition 2.1 ([1]). Let x, x′ ∈ X, y, y′ ∈ Y . Then (x, y) and (x′, y′) are
NP(κ, λ)-adjacent in X × Y if and only if

• x = x′ and y and y′ are λ-adjacent; or
• x and x′ are κ-adjacent and y = y′; or
• x and x′ are κ-adjacent and y and y′ are λ-adjacent. �

2.2. Connectedness. A subset Y of a digital image (X, κ) is κ-connected [32],
or connected when κ is understood, if for every pair of points a, b ∈ Y there
exists a sequence {yi}

m
i=0 ⊂ Y such that a = y0, b = ym, and yi and yi+1 are

κ-adjacent for 0 ≤ i < m.
For two subsets A, B ⊂ X, we will say that A and B are adjacent when there

exist points a ∈ A and b ∈ B such that a and b are equal or adjacent. Thus
sets with nonempty intersection are automatically adjacent, while disjoint sets
may or may not be adjacent. It is easy to see that a finite union of connected
adjacent sets is connected.

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Generalized normal product adjacency in digital topology

2.3. Continuous functions. The following generalizes a definition of [32].

Definition 2.2 ([4]). Let (X, κ) and (Y, λ) be digital images. A function f :
X → Y is (κ, λ)-continuous if for every κ-connected A ⊂ X we have that f(A)
is a λ-connected subset of Y .

When the adjacency relations are understood, we will simply say that f is
continuous. Continuity can be reformulated in terms of adjacency of points:

Theorem 2.3 ([32, 4]). A function f : X → Y is continuous if and only
if, for any adjacent points x, x′ ∈ X, the points f(x) and f(x′) are equal or
adjacent. �

Note that similar notions appear in [18, 19] under the names immersion,
gradually varied operator, and gradually varied mapping.

Theorem 2.4 ([3, 4]). If f : (A, κ) → (B, λ) and g : (B, λ) → (C, µ) are
continuous, then g ◦ f : (A, κ) → (C, µ) is continuous. �

Example 2.5 ([32]). A constant function between digital images is continuous.
�

Example 2.6. The identity function 1X : (X, κ) → (X, κ) is continuous.

Proof. This is an immediate consequence of Theorem 2.3. �

Definition 2.7. Let (X, κ) be a digital image in Zn. Let x, y ∈ X. A κ-path
of length m from x to y is a set {xi}

m
i=0 ⊂ X such that x = x0, xm = y, and

xi−1 and xi are equal or κ-adjacent for 1 ≤ i ≤ m. If x = y, we say {x} is a
path of length 0 from x to x.

Notice that for a path from x to y as described above, the function f :
[0, m]Z → X defined by f(i) = xi is (c1, κ)-continuous. Such a function is also
called a κ-path of length m from x to y.

2.4. Digital homotopy. A homotopy between continuous functions may be
thought of as a continuous deformation of one of the functions into the other
over a finite time period.

Definition 2.8 ([4]; see also [26]). Let X and Y be digital images. Let f, g :
X → Y be (κ, κ′)-continuous functions. Suppose there is a positive integer m
and a function F : X × [0, m]Z → Y such that

• for all x ∈ X, F(x, 0) = f(x) and F(x, m) = g(x);
• for all x ∈ X, the induced function Fx : [0, m]Z → Y defined by

Fx(t) = F(x, t) for all t ∈ [0, m]Z

is (2, κ′)−continuous. That is, Fx(t) is a path in Y .
• for all t ∈ [0, m]Z, the induced function Ft : X → Y defined by

Ft(x) = F(x, t) for all x ∈ X

is (κ, κ′)−continuous.

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 403



Then F is a digital (κ, κ′)−homotopy between f and g, and f and g are digi-
tally (κ, κ′)−homotopic in Y . If for some x ∈ X we have F(x, t) = F(x, 0) for
all t ∈ [0, m]Z, we say F holds x fixed, and F is a pointed homotopy. ✷

We denote a pair of homotopic functions as described above by f ≃κ,κ′ g.
When the adjacency relations κ and κ′ are understood in context, we say
f and g are digitally homotopic (or just homotopic) to abbreviate “digitally
(κ, κ′)−homotopic in Y ,” and write f ≃ g.

Proposition 2.9 ([26, 4]). Digital homotopy is an equivalence relation among
digitally continuous functions f : X → Y . ✷

Definition 2.10 ([5]). Let f : X → Y be a (κ, κ′)-continuous function and let
g : Y → X be a (κ′, κ)-continuous function such that

f ◦ g ≃κ′,κ′ 1X and g ◦ f ≃κ,κ 1Y .

Then we say X and Y have the same (κ, κ′)-homotopy type and that X and
Y are (κ, κ′)-homotopy equivalent, denoted X ≃κ,κ′ Y or as X ≃ Y when κ
and κ′ are understood. If for some x0 ∈ X and y0 ∈ Y we have f(x0) = y0,
g(y0) = x0, and there exists a homotopy between f ◦ g and 1X that holds
x0 fixed, and a homotopy between g ◦ f and 1Y that holds y0 fixed, we say
(X, x0, κ) and (Y, y0, κ

′) are pointed homotopy equivalent and that (X, x0) and
(Y, y0) have the same pointed homotopy type, denoted (X, x0) ≃κ,κ′ (Y, y0) or
as (X, x0) ≃ (Y, y0) when κ and κ

′ are understood. ✷

It is easily seen, from Proposition 2.9, that having the same homotopy
type (respectively, the same pointed homotopy type) is an equivalence rela-
tion among digital images (respectively, among pointed digital images).

2.5. Connectivity preserving and continuous multivalued functions.
A multivalued function f : X → Y assigns a subset of Y to each point of x.
We will write f : X ⊸ Y . For A ⊂ X and a multivalued function f : X ⊸ Y ,
let f(A) =

⋃

x∈A f(x).

Definition 2.11 ([30]). A multivalued function f : X ⊸ Y is connectivity
preserving if f(A) ⊂ Y is connected whenever A ⊂ X is connected.

As is the case with Definition 2.2, we can reformulate connectivity preser-
vation in terms of adjacencies.

Theorem 2.12 ([15]). A multivalued function f : X ⊸ Y is connectivity
preserving if and only if the following are satisfied:

• For every x ∈ X, f(x) is a connected subset of Y .
• For any adjacent points x, x′ ∈ X, the sets f(x) and f(x′) are adjacent.

�

Definition 2.11 is related to a definition of multivalued continuity for subsets
of Zn given and explored by Escribano, Giraldo, and Sastre in [20, 21] based on
subdivisions. (These papers make a small error with respect to compositions,
that is corrected in [22].) Their definitions are as follows:

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Generalized normal product adjacency in digital topology

Definition 2.13. For any positive integer r, the r-th subdivision of Zn is

Z
n
r = {(z1/r, . . . , zn/r) | zi ∈ Z}.

An adjacency relation κ on Zn naturally induces an adjacency relation (which
we also call κ) on Znr as follows: (z1/r, . . . , zn/r), (z

′
1/r, . . . , z

′
n/r) are adjacent

in Znr if and only if (z1, . . . , zn) and (z1, . . . , zn) are adjacent in Z
n.

Given a digital image (X, κ) ⊂ (Zn, κ), the r-th subdivision of X is

S(X, r) = {(x1, . . . , xn) ∈ Z
n
r | (⌊x1⌋, . . . , ⌊xn⌋) ∈ X}.

Let Er : S(X, r) → X be the natural map sending (x1, . . . , xn) ∈ S(X, r) to
(⌊x1⌋, . . . , ⌊xn⌋). �

Definition 2.14. For a digital image (X, κ) ⊂ (Zn, κ), a function f : S(X, r) →
Y induces a multivalued function F : X ⊸ Y if x ∈ X implies

F(x) =
⋃

x′∈E
−1
r (x)

{f(x′)}. �

Definition 2.15. A multivalued function F : X ⊸ Y is called continuous
when there is some r such that F is induced by some single valued continuous
function f : S(X, r) → Y . �

Figure 1. [15] Two images X and Y with their second sub-
divisions.

Note [15] that the subdivision construction (and thus the notion of continu-
ity) depends on the particular embedding of X as a subset of Zn. In particular
we may have X, Y ⊂ Zn with X isomorphic to Y but S(X, r) not isomorphic
to S(Y, r). This in fact is the case for the two images in Figure 1, when we
use 8-adjacency for all images. Then the spaces X and Y in the figure are iso-
morphic, each being a set of two adjacent points. But S(X, 2) and S(Y, 2) are
not isomorphic since S(X, 2) can be disconnected by removing a single point,
while this is impossible in S(Y, 2).

The definition of connectivity preservation makes no reference to X as being
embedded inside of any particular integer lattice Zn.

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 405



Proposition 2.16 ([20, 21]). Let F : X ⊸ Y be a continuous multivalued
function between digital images. Then

• for all x ∈ X, F(x) is connected; and
• for all connected subsets A of X, F(A) is connected. �

Theorem 2.17 ([15]). For (X, κ) ⊂ (Zn, κ), if F : X ⊸ Y is a continuous
multivalued function, then F is connectivity preserving. �

The subdivision machinery often makes it difficult to prove that a given
multivalued function is continuous. By contrast, many maps can easily be
shown to be connectivity preserving.

Proposition 2.18 ([15]). Let X and Y be digital images. Suppose Y is con-
nected. Then the multivalued function f : X ⊸ Y defined by f(x) = Y for all
x ∈ X is connectivity preserving. �

Proposition 2.19 ([15]). Let F : (X, κ) ⊸ (Y, λ) be a multivalued surjection
between digital images (X, κ), (Y, κ) ⊂ (Zn, κ). If X is finite and Y is infinite,
then F is not continuous. �

Corollary 2.20 ([15]). Let F : X ⊸ Y be the multivalued function between
digital images defined by F(x) = Y for all x ∈ X. If X is finite and Y is infinite
and connected, then F is connectivity preserving but not continuous. �

Examples of connectivity preserving but not continuous multivalued func-
tions on finite spaces are given in [15].

2.6. Other notions of multivalued continuity. Other notions of continu-
ity have been given for multivalued functions between graphs (equivalently,
between digital images). We have the following.

Definition 2.21 ([36]). Let F : X ⊸ Y be a multivalued function between
digital images.

• F has weak continuity if for each pair of adjacent x, y ∈ X, f(x) and
f(y) are adjacent subsets of Y .

• F has strong continuity if for each pair of adjacent x, y ∈ X, every
point of f(x) is adjacent or equal to some point of f(y) and every point
of f(y) is adjacent or equal to some point of f(x). �

Proposition 2.22 ([15]). Let F : X ⊸ Y be a multivalued function between
digital images. Then F is connectivity preserving if and only if F has weak
continuity and for all x ∈ X, F(x) is connected. �

Example 2.23 ([15]). If F : [0, 1]Z ⊸ [0, 2]Z is defined by F(0) = {0, 2},
F(1) = {1}, then F has both weak and strong continuity. Thus a multivalued
function between digital images that has weak or strong continuity need not
have connected point-images. By Theorem 2.12 and Proposition 2.16 it follows
that neither having weak continuity nor having strong continuity implies that a
multivalued function is connectivity preserving or continuous. ✷

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Generalized normal product adjacency in digital topology

Example 2.24 ([15]). Let F : [0, 1]Z ⊸ [0, 2]Z be defined by F(0) = {0, 1},
F(1) = {2}. Then F is continuous and has weak continuity but does not have
strong continuity. ✷

Proposition 2.25 ([15]). Let F : X ⊸ Y be a multivalued function between
digital images. If F has strong continuity and for each x ∈ X, F(x) is con-
nected, then F is connectivity preserving. �

The following shows that not requiring the images of points to be connected
yields topologically unsatisfying consequences for weak and strong continuity.

Example 2.26 ([15]). Let X and Y be nonempty digital images. Let the
multivalued function f : X ⊸ Y be defined by f(x) = Y for all x ∈ X.

• f has both weak and strong continuity.
• f is connectivity preserving if and only if Y is connected. �

As a specific example [15] consider X = {0} ⊂ Z and Y = {0, 2}, all with c1
adjacency. Then the function F : X ⊸ Y with F(0) = Y has both weak and
strong continuity, even though it maps a connected image surjectively onto a
disconnected image.

2.7. Shy maps and their inverses.

Definition 2.27 ([5]). Let f : X → Y be a continuous surjection of digital
images. We say f is shy if

• for each y ∈ Y , f−1(y) is connected, and
• for every y0, y1 ∈ Y such that y0 and y1 are adjacent, f

−1({y0, y1}) is
connected. �

Shy maps induce surjections on fundamental groups [5]. Some relation-
ships between shy maps f and their inverses f−1 as multivalued functions were
studied in [8, 15, 9]. We have the following.

Theorem 2.28 ([15, 9]). Let f : X → Y be a continuous surjection between
digital images. Then the following are equivalent.

• f is a shy map.
• For every connected Y0 ⊂ Y , f

−1(Y0) is a connected subset of X.
• f−1 : Y ⊸ X is a connectivity preserving multi-valued function.
• f−1 : Y ⊸ X is a multi-valued function with weak continuity such that
for all y ∈ Y , f−1(y) is a connected subset of X. �

2.8. Other tools. Other terminology we use includes the following. Given a
digital image (X, κ) ⊂ Zn and x ∈ X, the set of points adjacent to x ∈ Zn and
the neighborhood of x in Zn are, respectively,

Nκ(x) = {y ∈ Z
n | y is κ-adjacent to x},

N∗κ(x) = Nκ(x) ∪ {x}.

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 407



3. Extensions of normal product adjacency

In this section, we define extensions of the normal product adjacency, as
follows.

Definition 3.1. Let u and v be positive integers, 1 ≤ u ≤ v. Let {(Xi, κi)}
v
i=1

be digital images. Let NPu(κ1, . . . , κv) be the adjacency defined on the Carte-
sian product Πvi=1Xi as follows. For xi, x

′
i ∈ Xi, p = (x1, . . . , xv) and q =

(x′1, . . . , x
′
v) are NPu(κ1, . . . , κv)-adjacent if and only if

• For at least 1 and at most u indices i, xi and x
′
i are κi-adjacent, and

• for all other indices i, xi = x
′
i. �

Throughout this paper, the reader should be careful to note that some of
our results for NPu(κ1, . . . , κv) are stated for all u ∈ {1, . . . , v} and others are
stated only for u = 1 or u = 2 or u = v.

Proposition 3.2. NP(κ, λ) = NP2(κ, λ). I.e., given x, x
′ ∈ (X, κ) and y, y′ ∈

(Y, λ), p = (x, y) and p′ = (x′, y′) are NP(κ, λ)-adjacent in X × Y if and only
if p and p′ are NP2(κ, λ)-adjacent.

Proof. This follows immediately from Definitions 2.1 and 3.1. �

Theorem 3.3 ([13]). For X ∈ Zm, Y ∈ Zn, NP2(cm, cn) = cm+n, i.e.,
the normal product adjacency for (X, cm) × (Y, cn) coincides with the cm+n-
adjacency for X × Y . �

Examples are also given in [13] that show that if X ∈ Zm, Y ∈ Zn, and
a < m or b < n, then NP2(ca, cb) 6= ca+b.

The following shows that NPv obeys a recursive property.

Proposition 3.4. Let v > 2. Then

NPv(κ1, . . . , κv) = NP2(NPv−1(κ1, . . . , κv−1), κv).

Proof. Let xi, x
′
i ∈ Xi for 1 ≤ i ≤ v. Then p = (x1, . . . , xv) and p

′ =
(x′1, . . . , x

′
v) are NPv(κ1, . . . , κv)-adjacent if and only if for at least 1 and at

most v indices i, xi and x
′
i are κi-adjacent and for all other indices i, xi = x

′
i.

Hence p = (x1, . . . , xv) and p
′ = (x′1, . . . , x

′
v) are NPv(κ1, . . . , κu)-adjacent if

and only if either

• xi and x
′
i are κi-adjacent for from 1 to v−1 indices among {1, . . . , v−1},

xi = x
′
i for all other indices among {1, . . . , v − 1}, and xv = x

′
v; or

• xi and x
′
i are κi-adjacent for from 1 to v−1 indices among {1, . . . , v−1},

xi = x
′
i for all other indices among {1, . . . , v − 1}, and xv and x

′
v are

κv-adjacent.

Thus, p = (x1, . . . , xv) and p
′ = (x′1, . . . , x

′
v) are NPv(κ1, . . . , κv)-adjacent if

and only if p and p′ are NP2(NPu−1(κ1, . . . , κv−1), κv)-adjacent. �

Notice Proposition 3.4 may fail to extend to NPu(κ1, . . . , κv) if u < v, as
shown in the following (suggested by an example in [13]).

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Generalized normal product adjacency in digital topology

Example 3.5. Let xi, x
′
i ∈ (Xi, κi), i ∈ {1, 2, 3}. Suppose x1 and x

′
1 are

κ1-adjacent, x2 and x
′
2 are κ2-adjacent, and x3 = x

′
3. Then (x1, x2, x3) and

(x′1, x
′
2, x

′
3) are NP2(κ1, κ2, κ3)-adjacent in X1 × X2 × X3, but (x1, x2) and

(x′1, x
′
2) are not NP1(κ1, κ2)-adjacent in X1 × X2. Thus,

NP2(κ1, κ2, κ3) 6= NP2(NP1(κ1, κ2), κ3). �

Theorem 3.6. Let f, g : (X, κ) → (Y, λ) be functions. Let H : X×[0, m]Z → Y
be a function such that H(x, 0) = f(x) and H(x, m) = g(x) for all x ∈ X. Then
H is a homotopy if and only if H is (NP1(κ, c1), λ)-continuous.

Proof. In the following, we consider arbitrary (NP1(κ, c1), λ)-adjacent (x, t)
and (x′, t′) in X × [0, m]Z with x, x

′ ∈ X and t, t′ ∈ [0, m]Z. Such points offer
the following cases.

(1) x and x′ are κ-adjacent and t = t′; or
(2) x = x′ and t and t′ are c1-adjacent, i.e., |t − t

′| = 1.

Let H be a homotopy. Then f and g are continuous, and given (NP1(κ, c1), λ)-
adjacent (x, t) and (x′, t′) in X × [0, m]Z, we consider the cases listed above.

• In case 1, since H is a homotopy, H(x, t) and H(x′, t) = H(x′, t′) are
equal or λ-adjacent.

• In case 2, since H is a homotopy, H(x, t) and H(x′, t′) = H(x, t′) are
equal or λ-adjacent.

Therefore, H is (NP1(κ, c1), λ)-continuous.
Suppose H is (NP1(κ, c1), λ)-continuous. Then for κ-adjacent x, x

′, f(x) =
H(x, 0) and H(x′, 0) = f(x′) are equal or λ-adjacent, so f is continuous. Sim-
ilarly, g(x) = H(x, m) and g(x′) = H(x′, m) are equal or λ-adjacent, so g is
continuous. Also, the continuity of H implies that H(x, t) and H(x′, t) must
be equal or λ-adjacent, so the induced function Ht is (κ, λ)-continuous. For
c1-adjacent t, t

′, the continuity of H implies that H(x, t) and H(x, t′) are equal
or λ-adjacent, so the induced function Hx is continuous. By Definition 2.8, H
is a homotopy. �

4. NPu and maps on products

Given functions fi : (Xi, κi) → (Yi, λi), 1 ≤ i ≤ v, the function

Πvi=1fi : (Π
v
i=1Xi, NPu(κ1, . . . , κv)) → (Π

v
i=1Yi, NPu(λ1, . . . , λv))

is defined by

Πvi=1fi(x1, . . . , xv) = (f(x1), . . . , f(xv)), where xi ∈ Xi.

The following generalizes a result in [9, 13].

Theorem 4.1. Let fi : (Xi, κi) → (Yi, λi), 1 ≤ i ≤ v. Then the product map

f = Πvi=1fi : (Π
v
i=1Xi, NPv(κ1, . . . , κv)) → (Π

v
i=1Yi, NPv(λ1, . . . , λv))

is continuous if and only if each fi is continuous.

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 409



Proof. In the following, we let p = (x1, . . . , xv) and p
′ = (x′1, . . . , x

′
v), where

xi, x
′
i ∈ Xi.

Suppose each fi is continuous and p and p
′ are NPv(κ1, . . . , κv)-adjacent.

Then for all indices i, xi and x
′
i are equal or κi-adjacent, so fi(xi) and fi(x

′
i) are

equal or λi-adjacent. Therefore, f(p) and f(p
′) are equal or NPv(λ1, . . . , λv)-

adjacent. Thus, f is continuous.
Suppose f is continuous and for all indices i, xi and x

′
i are κi-adjacent. Then

f(p) and f(p′) are equal or NPv(λ1, . . . , λi)-adjacent. Therefore, for each index
i, fi(xi) and fi(x

′
i) are equal or λi-adjacent. Thus, each fi is continuous. �

The statement analogous to Theorem 4.1 is not generally true if cu-adjacencies
are used instead of normal product adjacencies, as shown in the following.

Example 4.2. Let X = {(0, 0), (1, 0)} ⊂ Z2. Let Y = {(0, 0), (1, 1)} ⊂ Z2.
Clearly, there is an isomorphism f : (X, c2) → (Y, c2). Consider X

′ = X ×
{0} ⊂ Z3 and Y ′ = Y × {0} ⊂ Z3. Note that the product map f × 1{0} is not
(c1, c1)-continuous, since X

′ is c1-connected and Y
′ = (f × 1{0})(X

′) is not
c1-connected. �

The following is a generalization of a result of [25].

Theorem 4.3. The projection maps pi : (Π
v
i=1Xi, NPu(κ1, . . . , κv)) → (Xi, κi)

defined by pi(x1, . . . , xv) = xi for xi ∈ (Xi, κi), are all continuous, for 1 ≤ u ≤
v.

Proof. Let p = (x1, . . . , xv) and p
′ = (x′1, . . . , x

′
v) be NPu(κ1, . . . , κv)-adjacent

in (Πvi=1Xi, NPu(κ1, . . . , κv)), where xi, x
′
i ∈ Xi. Then for all indices i, xi =

pi(p) and x
′
i = pi(p

′) are equal or κi-adjacent. Thus, pi is continuous. �

The statement analogous to Theorem 4.3 is not generally true if a cu-
adjacency is used instead of a normal product adjacency, as shown in the
following.

Example 4.4 ([13]). Let X = [0, 1]Z ⊂ Z. Let Y = {(0, 0), (1, 1)} ⊂ Z
2. Then

the projection map p2 : (X × Y, c3) → (Y, c1) is not continuous, since X × Y is
c3-connected and Y is not c1-connected. �

We see in the next result that isomorphism is preserved by taking Cartesian
products with a normal product adjacency.

Theorem 4.5. Let X = Πvi=1Xi. Let fi : (Xi, κi) → (Yi λi), 1 ≤ i ≤ v.

• For 1 ≤ u ≤ v, if the product map f = Πvi=1fi : (X, NPu(κ1, . . . , κv)) →
(Πvi=1Yi, NPu(λ1, . . . , κv)) is an isomorphism, then for 1 ≤ i ≤ v, fi is
an isomorphism.

• If fi is an isomorphism for all i, then the product map f = Π
v
i=1fi :

(X, NPv(κ1, . . . , κv)) → (Π
v
i=1Yi, NPv(λ1, . . . , κv)) is an isomorphism.

Proof. Let f be an isomorphism. Then each fi must be one-to-one and onto.
Let xi ∈ Xi. Let Ii : Xi → X be defined by

Ii(x) = (x1, . . . , xi−1, x, xi+1, . . . , xv).

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 410



Generalized normal product adjacency in digital topology

Define I′i : Yi → Y similarly. Clearly, Ii is (κi, NPu(κ1, . . . , κv)-continuous and
I′i is (λi, NPu(λ1, . . . , λv)-continuous. Let pi : X → Xi and p

′
i : Y → Yi be the

projections to the i-th coordinate. By Theorems 2.4 and 4.3, fi = p
′
i ◦ f ◦ Ii

and f−1i = pi ◦ f
−1 ◦ I′i are continuous. Hence, fi is an isomorphism.

Let fi : Xi → Yi be an isomorphism. One sees easily that f is one-to-one
and onto, and by Theorem 4.1, f is continuous. The inverse function f−1 is
the product function of the f−1i , hence is continuous by Theorem 4.1. Thus, f
is an isomorphism. �

The statement analogous to Theorem 4.5 is not generally true for all cu-
adjacencies, as shown by the following.

Example 4.6. Let X = {(0, 0), (1, 1)} ⊂ Z2. Let Y = {(0, 0), (1, 0)} ⊂ Z2.
Clearly, (X, c2) and (Y, c2) are isomorphic. Consider X

′ = X × {0} ⊂ Z3 and
Y ′ = Y × {0} ⊂ Z3. Note (X′, c1) and (Y

′, c1) are not isomorphic, since the
former is c1-disconnected and the latter is c1-connected. �

5. NPv and connectedness

Theorem 5.1. Let (Xi, κi) be digital images, i ∈ {1, 2, . . . , v}. Then (Xi, κi)
is connected for all i if and only (Πvi=1Xi, NPu(κ1, . . . , κv)) is connected.

Proof. Suppose (Xi, κi) is connected for all i. Let xi, x
′
i ∈ Xi. Then there are

paths Pi in Xi from xi to x
′
i. Let p = (x1, . . . , xv), p

′ = (x′1, . . . , x
′
v) ∈ Π

v
i=1Xi.

Then
⋃v

i=1 P
′
i , where

P ′1 = P1 × {(x2, . . . , xv)},

P ′i = {(x
′
1, . . . , x

′
i−1)} × Pi × {(xi+1, . . . , xv)} for 2 ≤ i < v,

P ′v = {(x
′
1, . . . , x

′
v−1)} × Pv,

is a path in Πvi=1Xi from p to p
′. Since p and p′ are arbitrarily selected points

in Πvi=1Xi, it follows that (Π
v
i=1Xi, NPu(κ1, . . . , κv)) is connected.

If (Πvi=1Xi, NPu(κ1, . . . , κv)) is connected, then (Xi, κi) = pi(Π
v
i=1Xi) is

connected, by Definition 2.2 and Theorem 4.3. �

The statement analogous to Theorem 5.1 is not generally true if a cu-
adjacency is used instead of NPu(κ1, . . . , κv) for X × Y , as shown by the
following.

Example 5.2 ([13]). Let X = [0, 1]Z, Y = {(0, 0), (1, 1)} ⊂ Z
2. Then X × Y

is c2-connected, but Y is not c1-connected. Also, X is c1-connected and Y is
c2-connected, but X × Y is not c1-connected. �

6. NPv and homotopy relations

In this section, we show that normal products preserve a variety of digital
homotopy relations. These include homotopy type and several generalizations
introduced in [17]. These generalizations - homotopic similarity, long homotopy
type, and real homotopy type - all coincide with homotopy type on pairs of finite
digital images; however, for each of these relationships, an example is given

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 411



in [17] of a pair of digital images, at least one member of which is infinite, such
that the two images have the given relation but are not homotopy equivalent.

By contrast with Euclidean topology, in which a bounded space such as a
single point and an unbounded space such as Rn with Euclidean topology can
have the same homotopy type, a finite digital image and an image with infinite
diameter - e.g., a single point and (Zn, c1) - cannot share the same homotopy
type. However, examples in [17] show that a finite digital image and an image
with infinite diameter can share homotopic similarity, long homotopy type, or
real homotopy type.

6.1. Homotopic maps and homotopy type.

Theorem 6.1. Let (Xi, κi) and (Yi, λi) be digital images, 1 ≤ i ≤ v. Let X =
(Πvi=1Xi, NPv(κ1, . . . , κi)). Let Y = (Π

v
i=1Yi, NPv(λ1, . . . , λi)). Let fi, gi :

Xi → Yi be continuous and let Hi : Xi ×[0, mi]Z → Yi be a homotopy from fi to
gi. Then there is a homotopy H between the product maps F = Π

v
i=1fi : X → Y

and G = Πvi=1gi : X → Y . If the homotopies Hi are pointed, then H is pointed.

Proof. Let M = max{mi}
v
i=1. Let H

′
i : Xi × [0, M]Z → Yi be defined by

H′i(x, t) =

{

Hi(x, t) for 0 ≤ t ≤ mi;
Hi(x, mi) for mi ≤ t ≤ M.

Clearly, H′i is a homotopy from fi to gi.
Let H : X × M → Y be defined by

H((x1, . . . , xv), t) = (H
′
1(x1, t), . . . , H

′
v(xv, t)).

It is easily seen that H is a homotopy from F to G, and that if each Hi is
pointed, then H is pointed. �

The following theorem generalizes results of [17].

Theorem 6.2. Suppose

(6.1) Xi ≃κi,λi Yi for 1 ≤ i ≤ v.

Then

(6.2) X = Πvi=1Xi ≃NPv(κ1,...,κv),NPv(λ1,...,λv) Y = Π
v
i=1Yi.

Further, if the homotopy equivalences (6.1) are all pointed with respect to xi ∈
Xi and yi ∈ Yi, then the homotopy equivalence (6.2) is pointed with respect to
(x1, . . . , xv) ∈ X and (y1, . . . , yv) ∈ Y .

Proof. We give a proof for the unpointed assertion. With minor modifications,
the pointed assertion is proven similarly.

By hypothesis, there exist continuous functions fi : Xi → Yi and gi : Yi →
Xi and homotopies Hi : Xi × [0, mi]Z → Xi from gi ◦ fi to 1Xi and Ki :
Yi × [0, ni]Z → Yi from fi ◦ gi to 1Yi.

Let M = max{mi}
v
i=1. Then H

′
i : Xi × [0, M]Z → Xi, defined by

H′i(x, t) =

{

Hi(x, t) for 0 ≤ t ≤ mi;
Hi(x, mi) for mi ≤ t ≤ M,

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 412



Generalized normal product adjacency in digital topology

is clearly a homotopy from gi ◦ fi to 1Xi.
Let F = Πvi=1fi : X → Y . Let G = Π

v
i=1gi : Y → X. By Theorem 4.1, F

and G are continuous. Let H : X × [0, M]Z → X be defined by

H(x1, . . . , xv, t) = (H
′
1(x1, t), . . . , H

′
v(xv, t)).

Then H is easily seen to be a homotopy from G ◦ F to Πvi=11Xi = 1X.
We can similarly show that F ◦ G ≃ 1Y . Therefore, X ≃ Y . �

The statements analogous to Theorems 6.1 and 6.2 are not generally true if
a cu-adjacency is used instead of a normal product adjacency for the Cartesian
product. Consider, e.g., X and Y as in Example 4.6. Let f : Y → X be a
(c1, c2)-isomorphism. Then f is (c1, c2)-homotopic to the constant map (0, 0)
of Y to (0, 0). However, f × 1{0} is not even (c1, c1)-continuous, hence is not

(c1, c1)-homotopic to (0, 0) × 1{0}. Although X and Y are (c2, c1)-homotopy
equivalent, X′ = X × {0} and Y ′ = Y × {0} are not (c1, c1)-homotopy equiva-
lent, since X′ is not c1-connected and Y

′ is c1-connected.

6.2. Homotopic similarity.

Definition 6.3 ([17]). Let X and Y be digital images. We say (X, κ) and (Y, λ)
are homotopically similar, denoted X ≃sκ,λ Y , if there exist subsets {Xj}

∞
j=1 of

X and {Yj}
∞
j=1 of Y such that:

• X =
⋃∞

j=1 Xj, Y =
⋃∞

j=1 Yj, and, for all j, Xj ⊂ Xj+1, Yj ⊂ Yj+1.
• There are continuous functions fj : Xj → Yj, gj : Yj → Xj such that
gj ◦ fj ≃κ,κ 1Xj and fj ◦ gj ≃λ,λ 1Yj .

• For m ≤ n, fn|Xm ≃κ,λ fm in Ym and gn|Ym ≃λ,κ gm in Xm.

If all of these homotopies are pointed with respect to some x1 ∈ X1 and
y1 ∈ Y1, we say (X, x1) and (Y, y1) are pointed homotopically similar, denoted
(X, x1) ≃

s
κ,λ (Y, y1) or (X, x1) ≃

s (Y, y1) when κ and λ are understood. ✷

Theorem 6.4. Let Xi ≃
s
κi,λi

Yi, 1 ≤ i ≤ v. Let X = Π
v
i=1Xi, X = Π

v
i=1Xi.

Then X ≃s
NPv(κ1,...,κv),NPv(λ1,...,λv)

Y . If the similarities Xi ≃
s
κi,λi

Yi are

pointed at xi ∈ Xi, yi ∈ Yi, then the similarity X ≃
s
NPv(κ1,...,κv),NPv(λ1,...,λv)

Y

is pointed at x0 = (x1, . . . , xv) ∈ X, y0 = (y1, . . . , yv) ∈ Y .

Proof. We give a proof for the unpointed assertion. A virtually identical argu-
ment can be given for the pointed assertion.

By hypothesis, for j ∈ N there exist digital images Xi,j ⊂ Xi, Yi,j ⊂ Yi
such that Xi,j ⊂ Xi,j+1, Xi =

⋃∞
j=1 Xi,j, Yi,j ⊂ Yi,j+1, Yi =

⋃∞
j=1 Yi,j, and

continuous functions fi,j : Xi,j → Yi,j, gi,j : Yi,j → Xi,j, such that gi,j ◦
fi,j ≃κi,κi 1Xi,j , fi,j ◦ gi,j ≃λi,λi 1Yi,j , and m ≤ n implies fi,n|Xi,m ≃κi,λi fi,m
in Yi,m and gi,n|Xi,m ≃λi,κi gi,m in Xi,m.

Let Xj = Π
v
i=1Xi,j, Yj = Π

v
i=1Yi,j. Clearly we have X =

⋃∞
j=1 Xj, Y =

⋃∞
j=1 Yj, Xj ⊂ Xj+1, Yj ⊂ Yj+1. Let fj = Π

v
i=1fi,j : Xj → Yj, gj =

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 413



Πvi=1gi,j : Yj → Xj. By Theorem 4.1, fj is (NPv(κ1, . . . , κv), NPv(λ1, . . . , λv))-
continuous and gj is (NPv(λ1, . . . , λv), NPv(κ1, . . . , κv))-continuous. By The-
orem 6.1,

gj◦fj ≃NPv(κ1,...,κv),NPv(κ1,...,κv) 1Xj and fj◦gj ≃NPv(λ1,...,λv),NPv(λ1,...,λv) 1Yj .

Also by Theorem 6.1, m ≤ n implies fn|Xm ≃NPv(κ1,...,κv),NPv(λ1,...,λv) fm
in Ym and gn|Ym ≃NPv(λ1,...,λv),NPv(κ1,...,κv) gm in Xm. This completes the
proof. �

6.3. Long homotopy type.

Definition 6.5 ([17]). Let (X, κ) and (Y, λ) be digital images. Let f, g : X → Y
be continuous. Let F : X × Z → Y be a function such that

• for all x ∈ X, there exists NF,x ∈ N such that t ≤ −NF,x implies
F(x, t) = f(x) and t ≥ NF,x implies F(x, t) = g(x).

• For all x ∈ X, the induced function Fx : Z → Y defined by

Fx(t) = F(x, t) for all t ∈ Z

is (c1, λ)-continuous.
• For all t ∈ Z, the induced function Ft : X → Y defined by

Ft(x) = F(x, t) for all x ∈ X

is (κ, λ)-continuous.

Then F is a long homotopy from f to g. If for some x0 ∈ X and y0 ∈ Y we
have F(x0, t) = y0 for all t ∈ N

∗, we say F is a pointed long homotopy. We
write f ≃L

κ,λ
g, or f ≃L g when the adjacencies κ and λ are understood, to

indicate that f and g are long homotopic functions. ✷

We have the following.

Theorem 6.6. Let fi, gi : (Xi, κi) → (Yi, λi) be continuous functions that
are long homotopic, 1 ≤ i ≤ v. Then f = Πvi=1fi and g = Π

v
i=1gi are long

homotopic maps from (Πvi=1Xi, NPv(κ1, . . . , κv)) to (Π
v
i=1Yi, NPv(λ1, . . . , λv)).

If the long homotopies fi ≃
L gi are pointed with respect to xi ∈ Xi and yi ∈ Yi,

then the long homotopy f ≃L g is pointed with respect to (x1, . . . , xv) ∈ Π
v
i=1Xi

and (y1, . . . , yv) ∈ Π
v
i=1Yi.

Proof. We give a proof for the unpointed assertion. Minor modifications yield
a proof for the pointed assertion.

Let hi : Xi × Z → Yi be a long homotopy from fi to gi. For all xi ∈ Xi,
there exists NFi,xi ∈ N such that t ≤ −NFi,xi implies hi(xi, t) = fi(xi) and
t ≥ NFi,xi implies hi(xi, t) = gi(xi). For all x = (x1, . . . , xv) ∈ Π

v
i=1Xi, let

Nx = max{NFi,xi | 1 ≤ i ≤ v}. Let h = Π
v
i=1hi : Π

v
i=1Xi × Z → Π

v
i=1Yi. Then

t ≤ −Nx implies h(x, t) = f(x) and t ≥ Nx implies h(x, t) = g(x).
For all x ∈ Πvi=1Xi, the induced function hx(t) = (hi(x1, t), . . . , hv(xv, t)) is

(c1, NPv(λ1, . . . , λv))-continuous, by an argument similar to that given in the
proof of Theorem 6.1.

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 414



Generalized normal product adjacency in digital topology

For all t ∈ Z, the induced function ht(x) = (hi(x1, t), . . . , hv(xv, t)) is
(NPv(κ1, . . . , κv), NPv(λ1, . . . , λv))-continuous, by an argument similar to that
given in the proof of Theorem 6.1. The assertion follows. �

Definition 6.7 ([17]). Let f : (X, κ) → (Y, λ) and g : (Y, λ) → (X, κ) be
continuous functions. Suppose g◦f ≃L 1X and f◦g ≃

L 1Y . Then we say (X, κ)
and (Y, λ) have the same long homotopy type, denoted X ≃L

κ,λ
Y or simply

X ≃L Y . If there exist x0 ∈ X and y0 ∈ Y such that f(x0) = y0, g(y0) = x0,
the long homotopy g ◦ f ≃L 1X holds x0 fixed, and the long homotopy f ◦ g ≃

L

1Y holds y0 fixed, then (X, x0, κ) and (Y, y0, λ) have the same pointed long
homotopy type, denoted (X, x0) ≃

L
κ,λ

(Y, y0) or (X, x0) ≃
L (Y, y0). ✷

Theorem 6.8. Let Xi ≃
L
κi,λi

Yi, 1 ≤ i ≤ v. Let X = Π
v
i=1Xi, Y = Π

v
i=1Yi.

Then X ≃L
NPv(κ1,...,κv),NPv(λ1,...,λv)

Y . If for each i the long homotopy equiva-

lence Xi ≃
L
κi,λi

Yi is pointed with respect to xi ∈ Xi and yi ∈ Yi, then the long

homotopy equivalence X ≃L
NPv(κ1,...,κv),NPv(λ1,...,λv)

Y is pointed with respect

to x0 = (x1, . . . , xv) ∈ X and y0 = (y1, . . . , yv) ∈ Y .

Proof. This follows easily from Definition 6.7 and Theorem 6.6. �

6.4. Real homotopy type.

Definition 6.9 ([17]). Let (X, κ) be a digital image, and [0, 1] ⊂ R be the unit
interval. A function f : [0, 1] → X is a real [digital] [κ-]path in X if:

• there exists ǫ0 > 0 such that f is constant on (0, ǫ0) with constant value
equal or κ-adjacent to f(0), and

• for each t ∈ (0, 1) there exists ǫt > 0 such that f is constant on each
of the intervals (t − ǫt, t) and (t, t + ǫt), and these two constant values
are equal or κ-adjacent, with at least one of them equal to f(t), and

• there exists ǫ1 > 0 such that f is constant on (1 − ǫ1, 1) with constant
value equal or κ-adjacent to f(1).

If t = 0 and f(0) 6= f((0, ǫ0)), or 0 < t < 1 and the two constant values
f((t − ǫt, t)) and f((t, t + ǫt)) are not equal, or t = 1 and f(1) 6= f((1 − ǫ1, 1)),
we say t is a jump of f.

Proposition 6.10 ([17]). Let p, q ∈ (X, κ). Let f : [a, b] → X be a real κ-path
from p to q. Then the number of jumps of f is finite.

Definition 6.11 ([17]). Let (X, κ) and (Y, κ′) be digital images, and let f, g :
X → Y be (κ, κ′) continuous. Then a real [digital] homotopy of f and g is a
function F : X × [0, 1] → Y such that:

• for all x ∈ X, F(x, 0) = f(x) and F(x, 1) = g(x)
• for all x ∈ X, the induced function Fx : [0, 1] → Y defined by

Fx(t) = F(x, t) for all t ∈ [0, 1]

is a real κ-path in X.

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 415



• for all t ∈ [0, 1], the induced function Ft : X → Y defined by

Ft(x) = F(x, t) for all x ∈ X

is (κ, κ′)–continuous.

If such a function exists we say f and g are real homotopic and write f ≃R g.
If there are points x0 ∈ X and y0 ∈ Y such that F(x0, t) = y0 for all t ∈ [0, 1],
we say f and g are pointed real homotopic.

Definition 6.12 ([17]). We say digital images (X, κ) and (Y, κ′) have the
same real homotopy type, denoted X ≃Rκ,κ′ Y or X ≃

R Y when κ and κ′ are
understood, if there are continuous functions f : X → Y and g : Y → X such
that g ◦ f ≃R 1X and f ◦ g ≃

R 1Y . If there exist x0 ∈ X and y0 ∈ Y such
that f(x0) = y0, g(y0) = x0, and the real homotopies above are pointed with
respect to x0 and y0, we say X and Y have the same pointed real homotopy
type, denoted (X, x0) ≃

R

κ,κ′ (Y, y0) or (X, x0) ≃
R (Y, y0).

Theorem 6.13. Suppose

(6.3) Xi ≃
R

κi,λi
Yi for 1 ≤ i ≤ v.

Let X = Πvi=1Xi, Y = Π
v
i=1Yi. Then

(6.4) X ≃RNPv(κ1,...,κv),NPv(λ1,...,λv) Y.

If the equivalences (6.3) are all pointed with respect to xi ∈ Xi and yi ∈ Yi,
then the equivalence (6.4) is pointed with respect to x0 = (x1, . . . , xv) ∈ X and
y0 = (y1, . . . , yv) ∈ Y .

Proof. We give a proof for the unpointed assertion. With minor modifications,
the same argument yields the pointed assertion.

By hypothesis, there exist continuous functions fi : Xi → Yi, gi : Yi → Xi
and real homotopies hi : Xi×[0, 1] → Xi from gi ◦fi to 1Xi, ki : Yi ×[0, 1] → Xi
from fi ◦ gi to 1Yi.

Let f = Πvi=1fi : X → Y . Let g = Π
v
i=1gi : Y → X. For x = (x1, . . . , xv) ∈

X with xi ∈ Xi, let H : X × [0, 1] → X be defined by

H(x, t) = (h1(x1, t), . . . , hv(xv, t)).

Then H(x, 0) = G ◦ F(x) and H(x, 1) = x.
For x ∈ X, the induced function Hx has jumps only at the finitely many

(by Proposition 6.10) jumps of the functions hi. It follows that Hx is a real
NPv(λ1, . . . , λv)-path in Y .

Let x′ = (x′1, . . . , x
′
v) be NPv(κ1, . . . , κv)-adjacent to x in X. Then, for any

t ∈ [0, 1], Ht(x
′) = (H1(x

′
1, t), . . . , Hv(x

′
v, t)) is NPv(λ1, . . . , λv)-adjacent to

Ht(x) = (H1(x1, t), . . . , Hv(xv, t)), since each Hi is a real homotopy. Hence Ht
is continuous.

Thus, H is a real homotopy from G ◦ F to 1X. A similar argument lets us
conclude that F ◦ G ≃R 1Y . Therefore, X ≃

R Y . �

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 416



Generalized normal product adjacency in digital topology

7. NPv and retractions

Definition 7.1 ([2, 3]). Let Y ⊂ (X, κ). A (κ, κ)-continuous function r : X →
Y is a retraction, and A is a retract of X, if r(y) = y for all y ∈ Y . �

Theorem 7.2. Let Ai ⊂ (Xi, κi), i ∈ {1, . . . , v}. Then Ai is a retract of Xi
for all i if and only if Πvi=1Ai is a retract of (Π

v
i=1Xi, NPv(κ1, . . . , κv)).

Proof. Suppose, for all i, Ai is a retract of Xi. Let ri : Xi → Ai be a retraction.
Then, by Theorem 4.1, Πvi=1ri : Π

v
i=1Xi → Π

v
i=1Ai is continuous, and therefore

is easily seen to be a retraction.
Suppose there is a retraction r : Πvi=1Xi → Π

v
i=1Ai. We construct retrac-

tions rj : Xj → Aj as follows. Let ai ∈ Ai. Define Ij : Xj → Π
v
i=1Xi by

Ij(x) = (a1, . . . , aj−1, x, aj+1, . . . , av).

Clearly, Ij is continuous. Then rj = pj ◦ r ◦ Ij is continuous, by Theorem 2.4
and Corollary 4.3, and is easily seen to be a retraction. �

Let A ⊂ (X, κ). We say A is a deformation retract of X if there is a
κ-homotopy H : X × [0, m]Z → X from 1X to a retraction of X to A. If
H(a, t) = a for all (a, t) ∈ Y × [0, m]Z, we say H is a strong deformation and A
is a strong deformation retract of X. We have the following.

Theorem 7.3. Let Ai ⊂ (Xi, κi), i ∈ {1, . . . , v}. Then Ai is a (strong) defor-
mation retract of Xi for all i if and only if A = Π

v
i=1Ai is a (strong) deforma-

tion retract of X = (Πui=1Xi, NPv(κ1, . . . , κv)).

Proof. Suppose Ai is a deformation retract of Xi, 1 ≤ i ≤ v. It follows from
Theorems 6.1 and 7.2 and that A is a deformation retract of X. If each Ai is
a strong deformation retract of Xi, then by using the argument in the proof of
Theorem 6.1 we can construct a homotopy from 1X to a retraction of X to A
that holds every point of a fixed, so A is a strong deformation retract of X.

Suppose A is a (strong) deformation retract of X. This means there is a
homotopy H : X×[0, m]Z → X from 1X to a retraction r of X onto A (such that
H(a, t) = a for all (a, t) ∈ A × [0, m]Z). Let Ii : Xi → X be as in Theorem 7.2.
Let Hi : Xi × [0, m]Z → Xi be defined by Hi(x, t) = pi(H(Ii(x), t)). Then Hi
is a homotopy between pi ◦ Ii = 1Xi and pi ◦ r ◦ Ii (such that Hi(ai, t) = ai for
all ai ∈ Ai). Since

pi ◦ r ◦ Ii(Xj) ⊂ pi ◦ r(X) = pi(A) = Ai

and a ∈ Ai implies pi ◦ r ◦ Ii(a) = a, pi ◦ r ◦ Ii is a retraction. Thus, Ai is a
(strong) deformation retract of Xi. �

8. NPv and the digital Borsuk-Ulam theorem

The Borsuk-Ulam Theorem of Euclidean topology states that if f : Sn → Rn

is a continuous function, where Rn is n-dimensional Euclidean space and Sn is

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 417



the unit sphere in Rn+1, i.e.,

Sn = {(x1, . . . , xn+1) ∈ R
n+1 |

n+1
∑

i=1

x2i = 1},

then there exists x ∈ Sn such that f(−x) = f(x). A “layman’s example” of
this theorem for n = 2 is that there are opposite points x, −x on the earth’s
surface with the same temperature and the same barometric pressure.

We say a set X ⊂ Zn is symmetric with respect to the origin if for every
x ∈ X, −x ∈ X.

The assertion analogous to the Borsuk-Ulam theorem is not generally true in
digital topology. An example is given in [7] of a continuous function f : S → Z
from a simple closed curve S ⊂ (Z2, c2), a digital analog of S

1, into the digital
line Z, with S symmetric with respect to the origin, such that f(x) 6= f(−x) for
all x ∈ S. However, the papers [7, 34] give conditions under which a continuous
function f from a digital version Sn of S

n to Zn must have a point x ∈ Sn for
which f(x) and f(−x) are equal or adjacent.

In particular, [34] uses the “boundary” of a digital box as a digital model of
a Euclidean sphere. Let Bn = Π

n
i=1[−ai, ai]Z, for ai ∈ N. Let

δBn =

n
⋃

i=1

{(x1, . . . , xn) ∈ Bn | xi ∈ {−ai, ai}}.

Theorem 8.1. We have the following.

• [7] Let (S, κ) be a digital simple closed curve in Zn such that S is
symmetric with respect to the origin. Let f : S → Z be a (κ, c1)-
continuous function. Then for some x ∈ S, f(x) and f(−x) are equal
or c1-adjacent, i.e., |f(x) − f(−x)| ≤ 1.

• [34] Let u ∈ {1, n − 1} and letf : δBn → Z
n−1 be a (cn, cu)-continuous

function. Then for some x ∈ δBn, f(x) and f(−x) are equal or cu-
adjacent. �

Notice that for Xi ⊂ Z
ni, Πvi=1Xi is symmetric with respect to the origin of

Z

∑
v
i=1

ni if and only if Xi is symmetric with respect to the origin of Z
ni for all

indices i.
Suppose m, n ∈ N, 1 ≤ m ≤ n. Let’s say a digital image S ⊂ Zn+1 that is

symmetric with respect to the origin has the (m, κ, λ)-Borsuk-Ulam property if
for every (κ, λ)-continuous function f : S → Zm there exists x ∈ X such that
f(x) and f(−x) are equal or λ-adjacent in Zn.

We have the following.

Theorem 8.2. Suppose

• v > 1,
• Si ⊂ Z

ni+1 is symmetric with respect to the origin of Zni+1 for 1 ≤
i ≤ v, and

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 418



Generalized normal product adjacency in digital topology

• Πvi=1Si has the (m, NPv(κ1, . . . , κv), NPv(λ1, . . . , λv))-Borsuk-Ulam prop-
erty for some adjacencies κi for Z

ni+1 and λi for Z
ni, where m =

∑v
i=1 ni.

Then, for all i, Si has the (ni, κi, λi)-Borsuk-Ulam property.

Proof. Notice Πvi=1Si ⊂ Z
m+v. Let f : Πvi=1Si → Z

m be any function that
is (NPv(κ1, . . . , κv), NPv(λ1, . . . , λv))-continuous. By hypothesis, there exists
x ∈ Πvi=1Si such that f(x) and f(−x) are equal or NPv(λ1, . . . , λv)-adjacent.

In particular, we can let f be the product of arbitrary continuous functions
fi : Si → Z

ni, since if fi : Si → Z
ni is (κi, λi)-continuous, then by Theorem 4.1,

f = Πvi=1fi is (NPv(κ1, . . . , κv), NPv(λ1, . . . , λv))-continuous. Therefore, there
exists x = (x1, . . . , xv) ∈ X where xi ∈ Si such that f(x) = (f1(x1), . . . , fv(xv))
and f(−x) = (f1(−x1), . . . , fv(−xv)) are equal or are NPv(λ1, . . . , λv)-adjacent.
Hence, for all indices i, fi(xi) and fi(−xi) are equal or λi-adjacent.

Since the fi were arbitrarily chosen, the assertion follows. �

9. NPu and the approximate fixed point property

In both topology and digital topology,

• a fixed point of a continuous function f : X → X is a point x ∈ X
satisfying f(x) = x;

• if every continuous f : X → X has a fixed point, then X has the fixed
point property (FPP).

However, a digital image X has the FPP if and only if X has a single point [10].
Therefore, it turns out that the approximate fixed point property is more inter-
esting for digital images.

Definition 9.1 ([10]). A digital image (X, κ) has the approximate fixed point
property (AFPP) if every continuous f : X → X has an approximate fixed
point, i.e., a point x ∈ X such that f(x) is equal or κ-adjacent to x. �

A number of results concerning the AFPP were presented in [10], including
the following.

Theorem 9.2 ([10]). Suppose (X, κ) has the AFPP. Let h : X → Y be a
(κ, λ)-isomorphism. Then (Y, λ) has the AFPP. �

Theorem 9.3 ([10]). Suppose Y is a retract of (X, κ). If (X, κ) has the AFPP,
then (Y, κ) has the AFPP. �

The following is a generalization of Theorem 5.10 of [10].

Theorem 9.4. Let (Xi, κi) be digital images, 1 ≤ i ≤ v. Then for any u ∈ Z
such that 1 ≤ u ≤ v, if (Πvi=1Xi, NPu(κ1, . . . , κv)) has the AFPP then (Xi, κi)
has the AFPP for all i.

Proof. Let X = (Πvi=1Xi, NPu(κ1, . . . , κv)).
Suppose X has the AFPP. Let xi ∈ Xi. Let

X′1 = X1 × {(x2, . . . , xv)},

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 419



X′i = {(x1, . . . , xi−1)} × Xi × {(xi+1, . . . , xv)} for 2 ≤ i < v,

X′v = {(x1, . . . , xv−1)} × XV .

Clearly, each X′i is a retract of X and is isomorphic to Xi. By Theorems 9.2
and 9.3, Xi has the AFPP. �

10. NPv and fundamental groups

Several versions of the fundamental group for digital images exist in the
literature, including those of [35, 27, 4, 16]. In this paper, we use the version
of [4], which was shown in [16] to be equivalent to the version developed in the
latter paper. Other papers cited in this section use the version of the digital
fundamental group presented in [4].

The author of [25] attempted to study the fundamental group of a Cartesian
product of digital simple closed curves. Errors of [25] were corrected in [13].

The notion of a covering map [25] is often useful in computing the fundamen-
tal group. The following is a somewhat simpler characterization of a covering
map than that given in [25].

Theorem 10.1 ([6]). Let (E, κ) and (B, λ) be digital images. Let g : E → B
be a (κ, λ)-continuous surjection. Then g is a (κ, λ)-covering map if and only
if for each b ∈ B, there is an index set M such that

• g−1(N∗λ(b, 1, B)) =
⋃

i∈M N
∗
κ(ei, 1, E) where ei ∈ g

−1(b);
• if i, j ∈ M and i 6= j then N∗κ(ei, 1, E) ∩ N

∗
κ(ej, 1, E) = ∅; and

• the restriction map g|N∗
κ
(ei,1,E) : N

∗
κ(ei, 1, E) → N

∗
λ
(b, 1, B) is a (κ, λ)-

isomorphism for all i ∈ M. �

Example 10.2 ([25]). Let C ⊂ Zn be a simple closed κ-curve, as realized by a
(c1, κ)-continuous surjection f : [0, m − 1]Z → C such that f(0) and f(m − 1)
are κ-adjacent. Define g : Z → C by g(z) = f(z mod m). Then g is a covering
map. �

Proposition 10.3 ([13]). Suppose for i ∈ {1, 2}, gi : Ei → Bi is a (κi, λi)-
covering map. Then g1×g2 : E1×E2 → B1×B2 is a (NP2(κ1, κ2), NP2(λ1, λ2))-
covering map. �

Corollary 10.4. Suppose for i ∈ {1, . . . , v}, gi : Ei → Bi is a (κi, λi)-covering
map. Then Πvi=1gi : Π

v
i=1Ei → Π

v
i=1Bi is a (NPv(κ1, . . . , κv), NPv(λ1, . . . , λv))-

covering map. �

Proof. This follows from Propositions 10.3 and 3.4, and Theorem 10.1. �

A digital image with the homotopy type of a single point is called con-
tractible. For the following theorem, it is useful to know that a digital simple
closed curve S is not contractible if and only if |S| > 4 [4, 7].

Theorem 10.5 ([29, 4, 25]). Let S ⊂ (Zn, κ) be a digital simple closed κ-
curve that is not contractible. Let s0 ∈ S. Then the fundamental group of S is
Πκ1(S, s0) ≈ Z. �

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 420



Generalized normal product adjacency in digital topology

The following theorem was discussed in [25], but the argument given for it
in [25] had errors. A correct proof was given in [13].

Theorem 10.6 ([13]). Let Si ⊂ (Z
ni, cni), for i ∈ {1, 2}, be a noncontractible

digital simple closed curve. Let si ∈ Si. Then the fundamental group

Π
cn1+n2
1 (S1 × S2, (s1, s2)) ≈ Z

2. �

�

The significance of the adjacency cn1+n2 in the proof of Theorem 10.6 is
that, per Theorem 3.3, NP(cn1, cn2) = cn1+n2. Thus, trivial modifications of
the proof given in [13] for Theorem 10.6 yield the following generalization.

Theorem 10.7. For i ∈ {1, . . . , v}, let Si ⊂ (Z
ni, κi) be a noncontractible

digital simple closed curve. Let si ∈ Si. Then the fundamental group

Π
NPv(κ1,...,κv)
1 (Π

v
i=1Si, (s1, . . . , sv)) ≈ Z

v. �

Many results concerning digital covering maps depend on the radius 2 local
isomorphism property (e.g., [24, 6, 11, 12, 13, 7, 14]). We have the following.

Definition 10.8 ([24]). Let n ∈ N. A (κ, λ)-covering (E, p, B) is a radius n
local isomorphism if, for all i ∈ M, the restriction map p|N∗κ(ei,n) : N

∗
κ(ei, n) →

N∗λ(bi, n) is an isomorphism, where ei, bi, M are as in Theorem 10.1.

Lemma 10.9. Let xi ∈ (Xi, κi). Then

N∗NPv(κ1,...,κv)((x1, . . . , xn), n) = Π
v
i=1N

∗
κi
(xi, n).

Proof. Let x = (x1, . . . , xv). Let y ∈ N
∗
NPv(κ1,...,κv)

(x, n). For some m ≤ n,

there is a path {yi}
m
i=0 from x to y. Let yi = (yi,1, . . . , yi,v) where yi,j ∈ Xi.

Since yi and yi+1 are NPv(κ1, . . . , κv)-adjacent, yi,j and yi,j+1 are equal or
κi-adjacent. Therefore, {yi,j}

m
j=1 is a κi path in Xi from yi,0 to yi,m. Hence

N∗
NPv(κ1,...,κv)

((x1, . . . , xn), n) ⊂ Π
v
i=1N

∗
κ1
(xi, n).

Let y = (y1, . . . , yv) ∈ Π
v
i=1N

∗
κ1
(xi, n). For each i and for some mi ≤ n,

there is a κi-path Pi = {yi,j}
mi
j=1 from xi to yi. There is no loss of generality in

assuming mi = n, since we can take Pi = {yi,j}
n
j=1 where yi,j = yi,mi for mi ≤

j ≤ n. Then for each i < n, y′i = (yi,1, . . . , yi,v) and y
′
i+1 = (yi+1,1, . . . , yi+1,v)

are equal or NPv(κ1, . . . , κv)-adjacent. Then {y
′
i}

n
i=1 is an NPv(κ1, . . . , κv)-

path from x to y. Thus, Πvi=1Nκ1(xi, n) ⊂ NNPv(κ1,...,κv)(x, n). The assertion
follows. �

Theorem 10.10. For 1 ≤ i ≤ v, let pi : (Ei, κi) → (Bi, λi) be continuous and
let n ∈ N. If (Ei, pi, Bi) is a covering and a radius n local isomorphism for
all i, then the product function

Πvi=1pi : Π
v
i=1Ei → Π

v
i=1Bi

is a (NPv(κ1, . . . , κv), NPv(λ1, . . . , λv)) covering map that is a radius n local
isomorphism.

Proof. This follows from Corollary 10.4 and Lemma 10.9. �

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 421



11. NPv and multivalued functions

We study properties of multivalued functions that are preserved by NPv.

11.1. Weak and strong continuity.

Theorem 11.1. Let Fi : (Xi, κi) ⊸ (Yi, λi) be multivalued functions for 1 ≤
i ≤ v. Let X = Πvi=1Xi, Y = Π

v
i=1Yi, and F = Π

v
i=1Fi : (X, NPv(κ1, . . . , κv)) ⊸

(Y, NPv(λ1, . . . , λv)). Then F has weak continuity if and only if each Fi has
weak continuity.

Proof. Let xi and x
′
i be κi-adjacent or equal in Xi. Then x = (x1, . . . , xv) and

x′ = (x′1, . . . , x
′
v) are NPv(κ1, . . . , κv)-adjacent or equal in X.

The multivalued function F has weak continuity ⇔ for x, x′ as above, F(x)
and F(x′) are NPv(λ1, . . . , λv)-adjacent subsets of Y ⇔ for each i and for all
xi, x

′
i as above, Fi(xi) and Fi(x

′
i) are λi-adjacent subsets of Yi ⇔ for each i, Fi

has weak continuity. �

Theorem 11.2. Let Fi : (Xi, κi) ⊸ (Yi, λi) be multivalued functions for 1 ≤
i ≤ v. Let X = Πvi=1Xi, Y = Π

v
i=1Yi, and F = Π

v
i=1Fi : (X, NPv(κ1, . . . , κv)) ⊸

(Y, NPv(λ1, . . . , λv)). Then F has strong continuity if and only if each Fi has
strong continuity.

Proof. Let xi and x
′
i be κi-adjacent or equal in Xi. Then x = (x1, . . . , xv) and

x′ = (x′1, . . . , x
′
v) are NPv(κ1, . . . , κv)-adjacent or equal in X.

The multivalued function F has strong continuity ⇔ for x, x′ as above, every
point of F(x) is NPv(λ1, . . . , λv)-adjacent or equal to a point of F(x

′) and every
point of F(x′) is NPv(λ1, . . . , λv)-adjacent or equal to a point of F(x) ⇔ for
each i and for all xi, x

′
i as above, every point of Fi(xi) is λi-adjacent or equal

to a point of Fi(x
′
i) and every point of Fi(x

′
i) is λi-adjacent or equal to a point

of Fi(xi) ⇔ for each i, Fi has strong continuity. �

11.2. Continuous multifunctions.

Lemma 11.3. Let X ⊂ Zm, Y ⊂ Zn. Let F : (X, ca) ⊸ (Y, cb) be a con-
tinuous multivalued function. Let f : (S(X, r), ca) → (Y, cb) be a continuous
function that induces F. Let s ∈ N. Then there is a continuous function
fs : (S(X, rs), ca) → (Y, cb) that induces F.

Proof. Given a point x = (x1, . . . , xm) ∈ S(X, rs), there is a unique point
I(x) = x′ = (x′1, . . . , x

′
m) ∈ S(X, r) such that x

′ “contains” x in the sense
that the fractional part of each component of x, xi − ⌊xi⌋, “truncates” to the
fractional part of the corresponding component of x′, x′i − ⌊x

′
i⌋, i.e.,

x′i − ⌊x
′
i⌋ ≤ xi − ⌊xi⌋ < x

′
i − ⌊x

′
i⌋ + 1/r.

(See Figure 2.) Define fs(x) = f(I(x)).
We must show fs is a continuous multivalued function that induces F . If

x, x′ are ca-adjacent in S(X, rs), then one sees easily that I(x) and I(x
′) are

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 422



Generalized normal product adjacency in digital topology

Figure 2. The digital image X = [0, 2]Z × [0, 1]Z with its
partitions S(X, 2) with member coordinates on heavy lines,
and S(X, 6) with member coordinates on both heavy and light
lines. In the notation used in the proof of Lemma 11.3, we
have, e.g., I(7/6, 2/3) = (1, 1/2).

equal or ca-adjacent in S(X, r). Hence fs(x) = f(I(x)) and fs(x
′) = f(I(x′))

are equal or cb-adjacent in Y . Thus, fs is continuous. For w ∈ X we have

F(w) =
⋃

y∈E
−1
r (w)

f(y) =
⋃

u∈E
−1
rs (w)

fs(u).

Therefore, f induces F . �

For multivalued functions Fi : (Xi, κi) ⊸ (Yi, λi), 1 ≤ i ≤ v, define the
product multivalued function

Πvi=1Fi : (Π
v
i=1Xi, NPv(κ1, . . . , κv)) ⊸ (Π

v
i=1Yi, NPv(λ1, . . . , λv))

by

(Πvi=1Fi)(x1, . . . , xv) = Π
v
i=1Fi(xi).

Theorem 11.4. Given multivalued functions Fi : (Xi, cai) ⊸ (Yi, cbi), 1 ≤
i ≤ v, if each Fi is continuous then the product multivalued function

Πvi=1Fi : (Π
v
i=1Xi, NPv(ca1, . . . , cav)) ⊸ (Π

v
i=1Yi, NPv(cb1, . . . , cbv ))

is continuous.

Proof. If each Fi is continuous, there exists a continuous fi : (S(Xi, ri), cai) →
(Yi, cbi) that generates Fi. By Lemma 11.3, we may assume that all the ri are

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 423



equal. Thus, for some positive integer r, we have fi : (S(Xi, r), cai) → (Yi, cbi)
generating Fi.

By Theorem 4.1, the product multivalued function

Πvi=1fi : (Π
v
i=1S(Xi, r), NPv(ca1, . . . , cav )) → (Π

v
i=1(Yi, NPv(cb1, . . . , cbv ))

is continuous. It is clear that this function generates the multivalued function
Πvi=1Fi. �

The paper [20] has several results concerning the following notions.

Definition 11.5 ([20]). Let (X, κ) ⊂ Zn be a digital image and Y ⊂ X. We
say that Y is a κ-retract of X if there exists a κ-continuous multivalued function
F : X ⊸ Y (a multivalued κ-retraction) such that F(y) = {y} if y ∈ Y . If
moreover F(x) ⊂ N∗cn(x) for every x ∈ X \ Y , we say that F is a multivalued
(N, κ)-retraction, and Y is a multivalued (N, κ)-retract of X.

We generalize Theorem 7.2 as follows.

Theorem 11.6. For 1 ≤ i ≤ v, let Ai ⊂ (Xi, κi) ⊂ Z
ni. Suppose Fi : Xi ⊸ Ai

is a continuous multivalued function for all i. Then Fi is a multivalued retrac-
tion for all i if and only if F = Πvi=1Fi : Π

v
i=1Xi ⊸ Π

v
i=1Ai is a multivalued

NPv(κ1, . . . , κv)-retraction. Further, Fi is an (N, κi)-retraction for all i if and
only if F is a multivalued (N, NPv(κ1, . . . , κv))-retraction.

Proof. Let X = Πvi=1Xi, A = Π
v
i=1Ai.

Suppose each Fi is a multivalued retraction. By Theorem 11.4, the product
multivalued function F is continuous.

Clearly, F(X) ⊂ A. Also, given a = (a1, . . . , av) ∈ A, we have

F(a) = Πvi=1Fi(ai) = Π
v
i=1{ai} = {a}.

Therefore, F(X) = A, and F is a multivalued retraction.
Conversely, suppose F is a multivalued retraction. By Theorem 11.4, each

Fi is continuous. Also, since F(X) = A, we must have, for each i, Fi(Xi) =
Ai, and since F is a retraction, Fi(a) = {a} for a ∈ Ai. Therefore, Fi is a
multivalued retraction.

Further, from Lemma 10.9, for x = (x1, . . . , xv) ∈ X, N
∗
NPv(cn1 ,...,cnv )

(x) =

Πvi=1Ncni (xi). It follows that Fi is an (N, κi)-retraction for all i if and only if

F is a multivalued (N, NPv(κ1, . . . , κv))-retraction. �

11.3. Connectivity preserving multifunctions.

Theorem 11.7. Let fi : (Xi, κi) ⊸ (Yi, λi) be a multivalued function between
digital images, 1 ≤ i ≤ v. Then the product map

Πvi=1fi : (Π
v
i=1Xi, NPv(κ1, . . . , κv)) ⊸ (Π

v
i=1Yi, NPv(λ1, . . . , λv))

is a connectivity preserving multifunction if and only if each fi is a connectivity
preserving multifunction.

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 424



Generalized normal product adjacency in digital topology

Proof. Let X = Πvi=1Xi, Y = Π
v
i=1Yi, F = Π

v
i=1fi : X ⊸ Y . Assume

x = (x1, . . . , xv), x
′ = (x′1, . . . , x

′
v)

with xi, x
′
i ∈ Xi. Using Theorem 2.12, we argue as follows.

F is connectivity preserving
⇔

• For every x ∈ X, F(x) = Πvi=1Fi(xi) is a connected subset of Y , and
• For adjacent x, x′ ∈ X, F(x) = Πvi=1Fi(xi) and F(x

′) = Πvi=1Fi(x
′
i)

are adjacent subsets of Y .

⇔

• For every xi ∈ Xi, Fi(x) is a connected subset of Yi, and
• For adjacent xi, x

′
i ∈ Xi, Fi(xi) and Fi(x

′
i) are adjacent subsets of Yi.

⇔
each Fi is connectivity preserving. �

12. NPv and shy maps

The following generalizes a result of [9].

Theorem 12.1. Let fi : (Xi, κi) → (Yi, λi) be a continuous surjection between
digital images, 1 ≤ i ≤ v. Then the product map

f = Πvi=1fi : (Π
v
i=1Xi, NPv(κ1, . . . , κv)) → (Π

v
i=1Yi, NPv(λ1, . . . , λv))

is shy if and only if each fi is a shy map.

Proof. Suppose the product map is shy. Since fi = pi ◦ f ◦ Ii, where Ii is the
continuous injection of the proof of Theorem 7.2, it follows from Theorems 2.4
and 4.3 that fi is continuous. Also, since f is surjective, fi must be surjective.

Let Y ′i be a λi-connected subset of Yi. By Theorem 5.1, Π
v
i=1Y

′
i is connected

in Πvi=1Yi. Since the product map is shy, we have from Theorem 2.28 that

X′ = f−1(Πvi=1Y
′
i ) = Π

v
i=1f

−1
i (Y

′
i )

is NPv(κ1, . . . , κv)-connected. Then f
−1
i (Y

′
i ) = pi(X

′) is κi-connected. From
Theorem 2.28, it follows that fi is shy.

Conversely, suppose each fi is shy. By Theorem 4.1, the product map Π
v
i=1fi

is continuous, and it is easily seen to be surjective.
Let yi ∈ Yi. Then (Π

v
i=1fi)

−1(y1, . . . , yv) = Π
v
i=1f

−1
i (yi) is connected, by

Definition 2.27 and Theorem 5.1.
Let yi, y

′
i be λi-adjacent in Yi, and let y = (y1, . . . , yv), y

′ = (y′1, . . . , y
′
v).

Then y and y′ are adjacent in Y , and (Πvi=1fi)
−1({y, y′}) = Πvi=1f

−1
i ({yi, y

′
i})

is connected, by Definition 2.27 and Theorem 5.1.
Thus, by Definition 2.27, Πvi=1fi is shy. �

The statement analogous to Theorem 12.1 is not generally true if cu-adjacencies
are used instead of normal product adjacencies, as shown in the following.

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 425



Example 12.2. Recall Example 4.2, in which X = {(0, 0), (1, 0)} ⊂ Z2, Y =
{(0, 0), (1, 1)} ⊂ Z2. There is a (c1, c2)-isomorphism f : X → Y . Consider
X′ = X × {0} ⊂ Z3, Y ′ = Y × {0} ⊂ Z3. Although the maps f and 1{0} are,
respectively, (c1, c2)- and (c1, c1)-isomorphisms and therefore are, respectively,
(c1, c2)- and (c1, c1)-shy, the product map f ×1{0} : X

′ → Y ′ is not (c1, c1)-shy,
by Theorem 2.28, since, as observed in Example 4.2, X′ is c1-connected and
Y ′ is not c1-connected. �

13. Further remarks

We have studied adjacencies that are extensions of the normal product adja-
cency for finite Cartesian products of digital images. We have shown that such
adjacencies preserve many properties for finite Cartesian products of digital
images that, in some cases, are not preserved by the use of the cu-adjacencies
most commonly used in the literature of digital topology.

Acknowledgements. We are grateful for the remarks of P. Christopher

Staecker, who suggested this study and several of its theorems.

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