Filomat 26:6 (2012), 1101–1114
DOI 10.2298/FIL1206101K
Published by Faculty of Sciences and Mathematics,
University of Ni
ˇ
s, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
Compression of Khalimsky topological spaces
Jeang Min Kang
a
, Sang-Eon Han
b
a
Department of Mathematics, Yonsei University, 262 Seongsanno, Seodaemun-Gu, Seoul, 126-749, Republic of Korea
b
Faculty of Liberal Education, Institute of Pure and Applied Mathematics,
Chonbuk National University, Jeonju-City Jeonbuk, 561-756, Republic of Korea
Abstract. Aiming at the study of the compression of Khalimsky topological spaces which is an interesting
field in digital geometry and computer science, the present paper develops a new homotopy thinning
suitable for the work. Since Khalimsky continuity of maps between Khalimsky topological spaces has
some limitations of performing a discrete geometric transformation, the paper uses another continuity (see
Definition 3.4) that can support the discrete geometric transformation and a homotopic thinning suitable
for studying Khalimsky topological spaces. By using this homotopy, we can develop a new homotopic
thinning for compressing the spaces and can write an algorithm for compressing 2D Khalimsky topological
spaces.
1. Introduction
Digital geometry is an approach to understanding and compressing qualitative properties of digital
images which have been studied in computer science, viewed as non-Hausdorff subspaces of Z
n
with some
particular choices of non-Hausdorff topology, n ∈ N, where Z
n
is the set of points in the Euclidean nD
space with integer coordinates and N represents the set of natural numbers. The idea of this subject is that
qualitative features of images are often in the center of interest in computer image processing, and further
that they provide a useful compression of images.
In digital geometry one of the interesting areas is the Khalimsky nD space which is a locally finite space
and satisfies the separation axiom T
0
instead of the Hausdorff separation axiom if n ≥ 2. In addition, the
Khalimsky 1D space satisfies the separation axiom T
1
[21]. Thus the present paper mainly studies subspaces
of the Khalimsky nD space from the viewpoint of digital geometry.
In relation to the study of discrete objects in Z
n
, we have used many tools from combinatorial topology,
graph theory, Khalimsky topology and so forth [7–9, 20, 21, 23, 24, 26, 30]. Motivated by Alexandroff
spaces in [1], the Khalimsky nD space, denoted by (Z
n
, T
n
), was established and its study includes the
papers [8, 13, 21, 28, 31]. Since the topology (Z
n
, T
n
) is established on the Euclidean nD space, it is useful to
consider a subset X ⊂ Z
n
to be a subspace of (Z
n
, T
n
) denoted by (X, T
n
X
), n ≥ 1 [1, 13].
2010 Mathematics Subject Classification. Primary 55P10; Secondary 55P15, 54A10, 65D18, 68U05, 68U10
Keywords. Khalimsky topological space, homotopy, k-homotopy, k-homotopic thinning, compression
Received: 18 October 2011; Revised: 06 June 2012; Accepted: 16 July 2012
Communicated by Ljubi
ˇ
sa D.R. Ko
ˇ
cinac
This research (the second author as a corresponding author) was supported by Basic Science Research Program through the
National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(2010002325).
This paper was supported by the selection of research-oriented professor of Chonbuk National University in 2011.
Email addresses: maniddo0002@naver.com (Jeang Min Kang), sehan@jbnu.ac.kr (Sang-Eon Han)
J.M. Kang, S.-E. Han / Filomat 26:6 (2012), 1101–1114 1102
In this regard, both a Khalimsky continuous map and a Khalimsky homeomorphism have been often
used in digital geometry [5, 6, 13, 18, 20–22, 24, 28, 31]. But it is well known that the graph k-connectivity
of Z
n
and every topological connectedness as well as Khalimsky connectedness are partially compatible
with each other [4, 5, 25]. Besides, a Khalimsky continuous map has some limitations of performing a
discrete geometric transformation such as a rotation by 90
◦
and a parallel translation with an odd vector
(see Remark 3.3). In order to overcome this difficulty, it can be helpful to take into account a reasonable
k-adjacency relations of Z
n
on (X, T
n
X
). Thus, considering a Khalimsky topological space (X, T
n
X
) with one
of the graph k-adjacency relations of (2.1), we call it a space if there is no danger of ambiguity and use the
notation (X, k, T
n
X
) := X
n,k
. The paper [13] introduced the category (briefly, CTC) consisting of both a set
Ob(CTC) of X
n,k
and a set Mor(X
n
0
,k
0
, Y
n
1
,k
1
) of (k
0
, k
1
)-continuous maps between each pair X
n
0
,k
0
and Y
n
1
,k
1
in Ob(CTC) (see Definition 3.4). The present paper uses the continuity in CTC (see Definition 3.4) instead
of both the Khalimsky continuity and the digital continuity in [3, 29] which leads to the development of a
homotopy suitable for studying Khalimsky topological spaces.
The paper proposes that an approach to the study of (X, T
n
X
) from the viewpoint of CTC can overcome
the limitation of Khalimsky continuity mentioned above. This is one of the reasons why we study a
Khalimsky topological space with graph k-connectivity and continuity of map in CTC. Finally, in relation
to the compression of Khalimsky topological spaces in CTC, we can use a homotopy in CTC and develop a
homotopic thinning which can substantially contribute to the compression of the spaces X
n,k
.
This paper is organized as follows. Section 2 provides some basic notions. Section 3 compares a
Khalimksy continuous map with continuous maps in CTC and further, it refers to some utilities of the
category CTC. Section 4 develops a homotopy thinning in CTC and proposes a method of compressing
Khalimsky topological spaces in terms of the homotopic thinning in CTC. In addition it suggests an
algorithm for compressing spaces X
2,k
. Section 5 concludes the paper with a summary and a further work.
2. Preliminaries
Let us now review some basic notions and properties of Khalimsky nD spaces. Khalimsky topology
arises from the Khalimsky line. More precisely, Khalimsky line topology on Z is induced from the subbase
{[2n−1, 2n+1]
Z
: n ∈ Z } [1] (see also [21]). Namely, the family of the subset {{2n+1}, [2m−1, 2m+1]
Z
: m, n ∈ Z}
is a basis of the Khalimsky line topology on Z denoted by (Z, T). Indeed, Khalimsky line topology has
useful properties. For instance, the Khalimsky line (Z, T) is connected and if one point is removed, then it
consists of two components and is finally not connected [21], which is the similar property of the real line
with the usual topology (R, U), where R means the set of real numbers. Furthermore, the usual product
topology on Z
n
induced from (Z, T), denoted by (Z
n
, T
n
), is called the Khalimsky nD space. In the present
paper each space X ⊂ Z
n
will be considered to be a subspace (X, T
n
X
) induced from the Khalimsky nD space
(Z
n
, T
n
).
Let us recall basic terminology of the structure of (Z
n
, T
n
). A point x = (x
1
, x
2
, · · · , x
n
) ∈ Z
n
is called pure
open if all coordinates are odd, and pure closed if each of the coordinates is even [21] and the other points in
Z
n
is called mixed [21]. In each of the spaces of Figures 1, 2, 3, 4, 5, 6 and 7 a black big dot stands for a pure
open point and the symbols and • mean a pure closed point and a mixed point, respectively.
Since a Khalimsky continuous map f need not preserve the digital connectivity of Dom( f ), it is meaning-
ful to study a multi-dimensional Khalimsky topological space (X, T
X
) with k-connectivity denoted by X
n,k
. Thus
let us recall the digital k-connectivity of Z
n
, as follows.
As a generalization of the commonly used k-adjacency relations of Z
2
and Z
3
[26, 29], the k-adjacency
relations of Z
n
were represented in [7] (see also [9]) as follows.
For a natural number m with 1 ≤ m ≤ n, two distinct points
p = (p
1
, p
2
, · · · , p
n
) and q = (q
1
, q
2
, · · · , q
n
) ∈ Z
n
,
are k(m, n)-(briefly, k-)adjacent if
• there are at most m indices i such that | p
i
− q
i
| = 1 and
• for all other indices i such that | p
i
− q
i
| , 1, p
i
= q
i
.
J.M. Kang, S.-E. Han / Filomat 26:6 (2012), 1101–1114 1103
In this operator k := k(m, n) is the number of points q which are k-adjacent to a given point p according to
the numbers m and n in N, where “ :=” means equal by definition. Indeed, this k(m, n)-adjacency is another
presentation of the k-adjacency of [7, 9] (for more details, see [17]). Consequently, this operator leads to the
representation of the k-adjacency relations of Z
n
[16]:
k := k(m, n) =
n−1
∑
i=n−m
2
n−i
C
n
i
, (2.1)
where C
n
i
=
n!
n−i)! i!
.
For instance, 8-, 32-, 64- and 80-adjacency relations of Z
4
are considered and further, 10-, 50-, 130-, 210- and
242-adjacency relations of Z
5
are obtained.
Owing to the digital k-connectivity paradox [26], a set X ⊂ Z
n
with one of the k-adjacency relations of
Z
n
is usually considered in a quadruple (Z
n
, k,
¯
k, X), where n ∈ N, X ⊂ Z
n
is the set of points we regard
as belonging to the set depicted, k represents an adjacency relation for X and
¯
k represents an adjacency
relation for Z
n
− X, where k ,
¯
k except X ⊂ Z [29]. But the paper is not concerned with the
¯
k-adjacency
of X. We say that the pair (X, k) is a digital space with k-adjacency (briefly, (binary) digital space) in Z
n
and
a subset (X, k) of (Z
n
, k) is k-connected if it is not a union of two disjoint non-empty sets not k-adjacent to
each other [26]. In other words, for a set (X, k) in Z
n
, two distinct points x, y ∈ X are called k-connected if
there is a k-path f : [0, m]
Z
→ X whose image is a sequence (x
0
, x
1
, · · · , x
m
) consisting of the set of points
{ f (0) = x
0
= x, f (1) = x
1
, · · · , f (m) = x
m
= y} such that x
i
and x
i+1
are k-adjacent, i ∈ [0, m − 1]
Z
, m ≥ 1. The
number m is called the length of this k-path [26]. For a digital space (X, k) and a point x ∈ X, we say that
the maximal k-connected subset of (X, k) containing the point x ∈ X is the k-(connected) component of a point
x ∈ X [26]. For a digital graph connectivity k, a simple k-path in X is the sequence (x
i
)
i∈[0,m]
Z
such that x
i
and
x
j
are k-adjacent if and only if either j = i + 1 or i = j + 1 [26]. Further, a simple closed k-curve with l
elements in Z
n
, denoted by SC
n,l
k
[10], is the simple k -path (x
i
)
i∈[0,l−1]
Z
, where x
i
and x
j
are k-adjacent if and
only if j = i + 1(mod l) or i = j + 1(mod l) [26].
3. Comparison of a Khalimsky continuous map and continuous maps in CTC
In this section by comparing a Khalimsky continuous map with a continuous map in CTC, we refer
to some limitations of Khalimsky continuity of maps between Khalimsky topological spaces so that we
can speak out merits of a Khalimsky topological space with digital k-connectivity and justify the (k
0
, k
1
)-
continuity of Definition 3.4 which will be used in the paper. In the Khalimsky nD space (Z
n
, T
n
), as usual,
consider a subset X ⊂ Z
n
to be a subspace (X, T
n
X
) induced from (Z
n
, T
n
), where T
n
X
= {O ∩ X| O ∈ T
n
}.
In this paper we mainly study spaces (X, T
n
X
) with one of the k-adjacency relations of Z
n
that is denoted by
(X, k, T
n
X
) := X
n,k
[13] and is called a space. In relation to the establishment of various kinds of continuities of
maps between spaces X
n,k
[13], in digital geometry we have often used the following digital k-neighborhood,
denoted by N
k
(x, ε), [8] (see also [9]) based on the notions of both digital adjacency and a simple k-path in
Section 2.
Definition 3.1. ([8]; see also [9]) Let (X, k) be a digital space, X ⊂ Z
n
, x, y ∈ X, and ε ∈ N. By the digital
k-neighborhood N
k
(x, ε) we denote the set
{y ∈ X : l
k
(x, y) ≤ ε} ∪ {x},
where l
k
(x, y) is the length of a shortest simple k-path x to y in X. Besides, we put l
k
(x, y) = ∞ if there is no
k-path from x to y. Thus, if the k-component of x is the singleton {x}, then we assume that N
k
(x, ε) = {x} for
any ε ∈ N.
Consider digital spaces (X, k
0
) in Z
n
0
and (Y, k
1
) in Z
n
1
and further, a map f : (X, k
0
) → (Y, k
1
). Then the
digital continuity of f in [3] can be equivalently represented in this way [15]:
The map f is digitally (k
0
, k
1
)-continuous at a point x ∈ X if and only if
f (N
k
0
(x, 1)) ⊂ N
k
1
( f (x), 1). (3.1)
J.M. Kang, S.-E. Han / Filomat 26:6 (2012), 1101–1114 1104
By using the digital (k
0
, k
1
)-continuity of (3.1), we obtain a digital topological category, briefly DTC,
consisting of the following two sets [9] (see also [13]):
(1) A set of objects (X, k) in Z
n
;
(2) For every ordered pair of objects (X, k
0
) in Z
n
0
and (Y, k
1
) in Z
n
1
, a set of all digitally (k
0
, k
1
)-continuous
maps f : (X, k
0
) → (Y, k
1
) as morphisms.
In DTC, for {a, b} ⊂ Z with a b we assume [a, b]
Z
= {n ∈ Z| a ≤ n ≤ b} with 2-adjacency [26].
Let us now recall the Khalimksy topological k-neighborhood which can be used for establishing continuity
of maps between the spaces X
n,k
in CTC (see Definition 3.4).
Definition 3.2. ([8]; see also [13, 20]) Consider a space X
n,k
:= X, x, y ∈ X, and ε ∈ N.
(1) A subset V of X is called a Khalimsky topological neighborhood of x if there exists O
x
∈ T
n
X
such that
x ∈ O
x
⊆ V.
(2) If a digital k-neighborhood N
k
(x, ε) is a Khalimsky topological neighborhood of x in (X, T
n
X
), then this
set is called a Khalimsky topological k-neighborhood of x with radius ε and we use the notation N
∗
k
(x, ε)
instead of N
k
(x, ε).
In A
2,4
of Figure 1(a), no N
∗
4
(a
i
, 1) exists, i ∈ {0, 8} because the smallest open set containing the point
a
0
(resp, a
8
) is the set {a
11
, a
0
, a
1
, a
2
} (resp. {a
6
, a
7
, a
8
, a
9
}). In addition, we can obtain that N
∗
4
(a
0
, 2) =
{a
10
, a
11
, a
0
, a
1
, a
2
} and further, N
∗
4
(a
8
, 2) = {a
6
, a
7
, a
8
, a
9
, a
10
}.
Let us recall Khalimsky (briefly, K-)continuity of maps between Khalimsky topological spaces: As usual,
for two Khalimsky topological spaces (X, T
n
0
X
) := X and (Y, T
n
1
Y
) := Y a map f : X → Y is called continuous at
the point x ∈ X if for any o pen set O
f (x)
⊂ Y containing the point f (x) there is an open set O
x
⊂ X containing
the point x such that f(O
x
) ⊂ O
f (x)
. In terms of the Khalimsky continuity of the map f , we obtain the
Khalimsky topological category, briefly KTC, consisting of the following two sets [13]:
(1) A set of objects (X, T
n
X
);
(2) For every ordered pair of objects (X, T
n
0
X
) and (Y, T
n
1
Y
) a set of all Khalimsky (briefly, K-)continuous maps
f : (X, T
n
0
X
) → (Y, T
n
1
Y
) as morphisms.
a
2
a
1
a
a
(-2, 0) (0, 0)
3
(1, 3)
a
a
a
aa
a
6 4
0
a
5
11
7
9
a
8
A
2,4
10
(a)
0
c
1
c
3
c
2
c
SC
3,4
26
(b)
Figure 1: Existence of N
∗
k
(x, ε).
Hereafter, SC
n,l
k
established in Section 2 is assumed to be a subspace of (Z
n
, T
n
). As already mentioned
above, we can observe the following limitations of K-continuous maps in Mor(KT C).
Remark 3.3. (1) Let f : X
n
0
,k
0
→ Y
n
1
,k
1
be a K-continuous map. Then f need not map a k
0
-connected subset
into a k
1
-connected one [13] (see also the maps f and 1 in Figure 2(a) and (b) of the present paper). More
precisely, let us consider the two maps:
f : X
1
→ Y
1
and 1 : X
2
→ Y
2
in Figure 2(a) and (b), respectively. While they are K-continuous maps, f (resp.
1) cannot preserve the 4-connectivity (resp. the 8-connectivity).
J.M. Kang, S.-E. Han / Filomat 26:6 (2012), 1101–1114 1105
(2) Regard SC
n,l
k
:= (c
i
)
i∈[0,l−1]
Z
as a subspace induced from the Khalimsky nD space (Z
n
, T
n
) and consider
the self-map f : SC
n,l
k
→ SC
n,l
k
given by f (c
i
) = c
i+1(mod l)
. Then f need not be a K-continuous map.
More precisely, assume SC
3,4
26
:= (c
i
)
i∈[0,3]
Z
as a subspace of (Z
3
, T
3
) (see Figure 1(b)) and the self-map
f : SC
3,4
26
→ SC
3,4
26
given by f (c
i
) = c
i+1(mod 4)
. Then we can clearly observe that f cannot be a K-continuous
map because O
c
0
= {c
3
, c
0
, c
1
}, O
c
1
= {c
1
}, O
c
2
= {c
1
, c
2
, c
3
}, and O
c
3
= {c
3
}, where O
x
means the smallest open
set containing the point x.
(3) Let us consider the map f : (Z, T) → (Z, T) given by f (t) = t + 1 which is a parallel translation with
an odd vector. Then we can clearly observe that f cannot be a K-continuous map.
In view of Remark 3.3, to study spaces X
n,k
, we need to use another continuity that can support the
digital geometric transformation mentioned in Remark 3.3.
Definition 3.4. ([13]) For two spaces X
n
0
,k
0
:= X and Y
n
1
,k
1
:= Y a function f : X → Y is said to be (k
0
, k
1
)-
continuous at a point x ∈ X if f (N
∗
k
0
(x, r)) ⊂ N
∗
k
1
( f (x), s), where the number r is the least element of N such
that N
∗
k
0
(x, r) contains an open set including the point x and s is the least element of N such that N
∗
k
1
( f (x), s)
contains an open set including the point f (x).
Furthermore, we say that a map f : X → Y is (k
0
, k
1
)-continuous if the map f is (k
0
, k
1
)-continuous at every
point x ∈ X.
In Definition 3.4 if such a neighborhood N
∗
k
1
( f (x), ε) does not exist, then we clearly say that f cannot
be (k
0
, k
1
)-continuous at the point x. Further, in Definition 3.4 if k
0
= k
1
and n
0
= n
1
, then we call it a
k
0
-continuous map.
Let us now recall the category [13], denoted by CTC, consisting of the following two sets:
• A set of objects X
n,k
denoted by Ob(CTC);
• For every ordered pair of spaces X
n
0
,k
0
and Y
n
1
,k
1
in Ob(CTC) a set Mor(X
n
0
,k
0
, Y
n
1
,k
1
) of (k
0
, k
1
)-continuous
maps as morphisms.
In CTC, for {a, b} ⊂ Z with a b, [a, b]
Z
= {a ≤ n ≤ b} can be assumed to be a subspace of (Z, T) if it is
related to the Khalimsky topology (Z, T). Then ([a, b]
Z
, T
[a,b]
Z
) is called a Khalimsky interval [28] and is briefly
denoted by [a, b]
Z
. For a map f : X
n,k
→ Y
n,k
let us compare K-continuity of f in KTC with k-continuity of
f in CTC. Indeed, according to the dimension n and connectedness of X
n,k
and Y
n,k
, some intrinsic features
appear.
Theorem 3.5. Let f : X
1,2
→ Y
1,2
be a map. Then K-continuity of f in KTC implies 2-continuity of f in CTC. But
the converse does not hold.
Proof. In the Khalimsky line (Z, T) each point t ∈ Z has N
∗
2
(t, 1) = {t − 1, t, t + 1} ⊂ Z and further, the point t
is connected to the points t − 1 and t + 1. Thus a K-continuous map f : X
1,2
→ Y
1,2
implies a 2-continuous
map in CTC because a K-continuous map preserves connectedness under the Khalimsky line topology. Let
us now examine if the assertion is true or not with the following four cases at every point x ∈ X
1,2
and its
image f (x) := y ∈ Y
1,2
.
(Case 1) Assume that both x and y are pure closed points. Then a K-continuous map f : X
1,2
→ Y
1,2
is equivalent to a 2-continuous map in CTC because N
∗
2
(x, 1) = O
x
and N
∗
2
(y, 1) = O
y
, where O
t
means the
smallest open set containing t ∈ Z under the subspace topologies on X
1,2
and Y
1,2
.
(Case 2) Assume that both x and y are pure open points so that O
x
= {x} and O
y
= {y}. Since from the
hypothesis we obtain that f (O
x
) ⊂ O
y
, we clearly observe that f (N
∗
2
(x, 1)) ⊂ N
∗
2
(y, 1) because O
x
⊂ N
∗
2
(x, 1)
and O
y
⊂ N
∗
2
(y, 1) and further, if there is a point x
1
∈ N
∗
2
(x, 1) with x
1
, x such that f (x
1
) < N
∗
2
(y, 1), then
f cannot be a K-continuous map at the point x
1
because f does not preserve connectedness between the
points x
1
and x.
(Case 3) Assume that x is a pure closed point and y is a pure open point. Owing to both the hypothesis
of the K-continuity of f and the fact that N
∗
2
(x, 1) = O
x
, we obtain that f (N
∗
2
(x, 1)) ⊂ O
y
= {y}, which implies
that f (N
∗
2
(x, 1)) ⊂ N
∗
2
(y, 1) because O
y
⊂ N
∗
2
(y, 1).
