Existence of the Category DTC₂(K) Equivalent to the Given Category KAC₂
For a given category KAC₂, the present paper deals with an existence problem of the category DTC₂(k) which is equivalent to KAC₂, where DTC₂(k) is the category whose objects are simple closed k-curves with even number l of elements in Zⁿ, l ≠ 6 and morphisms are (digitally) k-continuous maps, and...
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irk-123456789-1664742020-02-23T01:25:42Z Existence of the Category DTC₂(K) Equivalent to the Given Category KAC₂ Han, S.-E. Статті For a given category KAC₂, the present paper deals with an existence problem of the category DTC₂(k) which is equivalent to KAC₂, where DTC₂(k) is the category whose objects are simple closed k-curves with even number l of elements in Zⁿ, l ≠ 6 and morphisms are (digitally) k-continuous maps, and KAC₂ is the category whose objects are simple closed A-curves and morphisms are A-maps. To address this issue, the paper starts with the category, denoted by KAC₁, whose objects are connected nD Khalimsky topological subspaces with Khalimsky adjacency and morphisms are A-maps in [Han S. E., Sostak A. A compression of digital images derived from a Khalimsky topological structure // Comput. and Appl. Math. – 2013. – 32. – P. 521 – 536]. Based on this approach, in KAC₁ the paper proposes the notions of an A-homotopy and an A-homotopy equivalence, and classifies spaces in KAC₁ or KAC₂ in terms of an A-homotopy equivalence. Finally, the paper proves that for a given category KAC₂ there is DTC₂(k) which is equivalent to KAC₂. Для заданої категорiї KAC₂ вивчено проблему iснування категорiї DTC₂(k), що еквiвалентна KAC₂, де DTC₂(k) — категорiя, об’єктами якої є простi замкненi k-кривi з парним числом l, l ≠ 6, елементiв в Zⁿ, а морфiзмами — (цифрово) k-неперервнi вiдображення, тодi як KAC₂ — категорiя, об’єктами якої є простi замкненi A-кривi, а морфiзми є A-вiдображеннями. Наш виклад ми починаємо з категорiї, що позначена KAC₁, об’єктами якої є nD зв’язнi топологiчнi пiдпростори Халiмського з сумiжнiстю Халiмського, а морфiзми є A-вiдображеннями, що визначенi в [Han S. E., Sostak A. A compression of digital images derived from a Khalimsky topological structure // Comput. and Appl. Math. – 2013. – 32. – P. 521 – 536]. На основi запропонованого пiдходу в категорiї KAC₁ введено поняття A-гомотопiї та A-гомотопiчної еквiвалентностi, а простори з KAC₁ або KAC₂ класифiковано в термiнах A-гомотопiчної еквiвалентностi. Насамкiнець доведено, що для заданої категорiї KAC₂ iснує DTC₂(k), еквiвалентнa KAC₂. 2015 Article Existence of the Category DTC₂(K) Equivalent to the Given Category KAC₂ / S.-E. Han // Український математичний журнал. — 2015. — Т. 67, № 8. — С. 1122–1133. — Бібліогр.: 10 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/166474 513.8 en Український математичний журнал Інститут математики НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
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Статті Статті |
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Статті Статті Han, S.-E. Existence of the Category DTC₂(K) Equivalent to the Given Category KAC₂ Український математичний журнал |
| description |
For a given category KAC₂, the present paper deals with an existence problem of the category DTC₂(k) which is
equivalent to KAC₂, where DTC₂(k) is the category whose objects are simple closed k-curves with even number l of
elements in Zⁿ, l ≠ 6 and morphisms are (digitally) k-continuous maps, and KAC₂ is the category whose objects are
simple closed A-curves and morphisms are A-maps. To address this issue, the paper starts with the category, denoted
by KAC₁, whose objects are connected nD Khalimsky topological subspaces with Khalimsky adjacency and morphisms
are A-maps in [Han S. E., Sostak A. A compression of digital images derived from a Khalimsky topological structure //
Comput. and Appl. Math. – 2013. – 32. – P. 521 – 536]. Based on this approach, in KAC₁ the paper proposes the notions
of an A-homotopy and an A-homotopy equivalence, and classifies spaces in KAC₁ or KAC₂ in terms of an A-homotopy
equivalence. Finally, the paper proves that for a given category KAC₂ there is DTC₂(k) which is equivalent to KAC₂. |
| format |
Article |
| author |
Han, S.-E. |
| author_facet |
Han, S.-E. |
| author_sort |
Han, S.-E. |
| title |
Existence of the Category DTC₂(K) Equivalent to the Given Category KAC₂ |
| title_short |
Existence of the Category DTC₂(K) Equivalent to the Given Category KAC₂ |
| title_full |
Existence of the Category DTC₂(K) Equivalent to the Given Category KAC₂ |
| title_fullStr |
Existence of the Category DTC₂(K) Equivalent to the Given Category KAC₂ |
| title_full_unstemmed |
Existence of the Category DTC₂(K) Equivalent to the Given Category KAC₂ |
| title_sort |
existence of the category dtc₂(k) equivalent to the given category kac₂ |
| publisher |
Інститут математики НАН України |
| publishDate |
2015 |
| topic_facet |
Статті |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/166474 |
| citation_txt |
Existence of the Category DTC₂(K) Equivalent to the Given Category KAC₂ / S.-E. Han // Український математичний журнал. — 2015. — Т. 67, № 8. — С. 1122–1133. — Бібліогр.: 10 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT hanse existenceofthecategorydtc2kequivalenttothegivencategorykac2 |
| first_indexed |
2025-07-14T21:49:16Z |
| last_indexed |
2025-07-14T21:49:16Z |
| _version_ |
1837660643051700224 |
| fulltext |
UDC 513.8
S.-E. Han (Inst. Pure and Appl. Math., Chonbuk Nat. Univ., Korea)
EXISTENCE OF THE CATEGORY DTC2(k) WHICH IS EQUIVALENT
TO THE GIVEN CATEGORY KAC2
ПРО IСНУВАННЯ КАТЕГОРIЇ DTC2(k), ЩО ЕКВIВАЛЕНТНА ЗАДАНIЙ
КАТЕГОРIЇ KAC2
For a given category KAC2, the present paper deals with an existence problem of the category DTC2(k) which is
equivalent to KAC2, where DTC2(k) is the category whose objects are simple closed k-curves with even number l of
elements in Zn, l 6= 6 and morphisms are (digitally) k-continuous maps, and KAC2 is the category whose objects are
simple closed A-curves and morphisms are A-maps. To address this issue, the paper starts with the category, denoted
by KAC1, whose objects are connected nD Khalimsky topological subspaces with Khalimsky adjacency and morphisms
are A-maps in [Han S. E., Sostak A. A compression of digital images derived from a Khalimsky topological structure //
Comput. and Appl. Math. – 2013. – 32. – P. 521 – 536]. Based on this approach, in KAC1 the paper proposes the notions
of an A-homotopy and an A-homotopy equivalence, and classifies spaces in KAC1 or KAC2 in terms of an A-homotopy
equivalence. Finally, the paper proves that for a given category KAC2 there is DTC2(k) which is equivalent to KAC2.
Для заданої категорiїKAC2 вивчено проблему iснування категорiїDTC2(k),що еквiвалентнаKAC2, деDTC2(k)—
категорiя, об’єктами якої є простi замкненi k-кривi з парним числом l, l 6= 6, елементiв в Zn, а морфiзмами —
(цифрово) k-неперервнi вiдображення, тодi як KAC2 — категорiя, об’єктами якої є простi замкненi A-кривi, а
морфiзми є A-вiдображеннями. Наш виклад ми починаємо з категорiї, що позначена KAC1, об’єктами якої є
nD зв’язнi топологiчнi пiдпростори Халiмського з сумiжнiстю Халiмського, а морфiзми є A-вiдображеннями, що
визначенi в [Han S. E., Sostak A. A compression of digital images derived from a Khalimsky topological structure
// Comput. and Appl. Math. – 2013. – 32. – P. 521 – 536]. На основi запропонованого пiдходу в категорiї KAC1
введено поняття A-гомотопiї та A-гомотопiчної еквiвалентностi, а простори з KAC1 або KAC2 класифiковано в
термiнах A-гомотопiчної еквiвалентностi. Насамкiнець доведено, що для заданої категорiї KAC2 iснує DTC2(k),
еквiвалентнa KAC2.
1. Introduction. Let Z, N and Zn represent the sets of integers, natural numbers and points in the
Euclidean nD space with integer coordinates, respectively. To recognize a set X ⊂ Zn with graph
theoretical structures, A. Rosenfeld introduced digital topology [19]. Furthermore, many researchers
have developed several tools such as a Marcuse Wyse topological structure [20], a graph theoretical
method [4 – 7, 16, 19], a Khalimsky topological structure [3, 10, 11, 14, 15, 18], a locally finite
topological approach [17] and so forth. Nowadays, digital topology plays an important role in some
areas of computer science and applied topology such as image processing, computer graphics, image
analysis, mathematical morphology and so forth. It has been used to study topological properties
and features, e.g., connectedness and boundaries of two, three or nD digital images. Since the paper
will frequently refer a Khalimsky topological structure, hereafter, for convenience we will use the
terminology K — instead of “Khalimsky” if there is no danger of ambiguity.
Since continuity of maps between digital spaces is also an important topic in digital topology,
many studies have examined various properties of a K-continuous map, connectedness, Khalimsky
adjacency, a K-homeomorphism [10, 11, 14, 15]. However, it turns out that a K-continuous map
have some limitations [13]: it does not contain some rotations and transformations. Thus the recent
paper [13] developed a broader class of maps, called A-maps, which are generalizations of both a
K-continuous map and a Khalimsky adjacency (for short, KA-) map. Furthermore, it introduced an
A-isomorphism which is a generalization of a K-homeomorphism. But we still need a mathematical
c© S.-E. HAN, 2015
1122 ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8
EXISTENCE OF THE CATEGORY DTC2(k) WHICH IS EQUIVALENT TO THE GIVEN CATEGORY KAC2 1123
structure associated with a K-topological structure which is equivalent to a Rosenfeld’s digital image
(X, k) in Zn. Let DTC be a category whose objects are digital images (X, k) in Zn and morphisms
are (digitally) k-continuous maps [6, 19]. Thus we see thatDTC is not a topological category because
these digital images in Ob(DTC) have only graph theoretical structures instead of topological ones.
Besides, let DTC1(k) (resp. DTC2(k)) be the category whose objects are k-connected digital images
(resp. simple closed k-curves with even number l of elements, l 6= 6) in Zn and morphisms are k-
continuous maps for the categories DTC1(k) and DTC2(k). It is obvious that both DTC1(k) and
DTC2(k) are subcategories of DTC. Let KAC1 (resp. KAC2) be the category whose objects are
connected K-topological spaces with K-adjacency (resp. simple closed A-curves) and morphisms
are A-maps for both KAC1 and KAC2. Let us now raise the following question:
Given the category KAC2, is there a category DTC2(k) which is equivalent to KAC2 ?
This problem is strongly related to the development of a homotopy associated with K-topology.
Up to now, there is no research into this construction from the viewpoint of K-topology. To address
this issue, the paper proposes an equivalence between KAC2 and DTC2(k) so that an A-map in
KAC2 is equivalent to a (digitally) k-continuous map in DTC2(k). Besides, in KAC1 the paper
establishes the notions of an A-homotopy and an A-homotopy equivalence (see Sections 5 and 6).
2. Preliminaries. Let us recall some basic facts and terminology from digital topology such as
Zn with digital k-connectivity and Khalimsky (for brevity, K-) topology. For two distinct points a
and b in Z let [a, b]Z = {n ∈ Z | a ≤ n ≤ b} [16]. The Khalimsky line topology on Z is generated
by the set {[2n− 1, 2n+ 1]Z : n ∈ Z} as a subbase [1] (see also [14, 15]). Furthermore, the product
topology on Zn induced by (Z, T ) is called the Khalimsky product topology on Zn (or Khalimsky
nD space) which is denoted by (Zn, Tn) [14]. In the present paper each space X ⊂ Zn related
to K-topology is considered to be a subspace (X,TnX) induced by (Zn, Tn). It is well known that
(Zn, Tn) is a T0-Alexandroff space [15] (cf. [1]).
Let us now recall the structure of (Zn, Tn). A point x = (x1, x2, . . . , xn) ∈ Zn is pure open if
all coordinates are odd; and it is pure closed if each of the coordinates is even [15]. The other points
in Zn are called mixed [15]. In each of the spaces of Figures 1, 2 and 3, a black jumbo dot means
a pure open point and further, the symbols � and • mean a pure closed point and a mixed point,
respectively. In relation to the further statement of a pure point and a mixed point, we say that a
point x is open if SN(x) = {x}, where SN(x) means the smallest neighborhood of x ∈ Zn. The
point x ∈ Zn is called closed if C(x) = {x}, where C(x) stands for a closure of the singleton {x}.
Many studies have investigated various properties of a K-continuous map, connectedness, Khal-
imsky adjacency, a K-homeomorphism [6, 8, 14, 15, 18]. It is important and well known that in
(Zn, Tn) for two distinct points an equivalence exists between connectedness and Khalimsky adja-
cency [15]. In (Zn, Tn), a Khalimsky adjacency relation is symmetric [15]. Besides, for any Khal-
imsky adjacent set A ⊂ X, the image by a K-continuous map f : (X,Tn0
X ) := X → (Y, Tn1
Y ) := Y,
f(A), is also a Khalimsky adjacent subset of Y, which is very useful properties for studying Khalim-
sky topological spaces. However, both a K-continuous map and a K-homeomorphism are very rigid
so that their contributions can be so limited. For instance, not every rotation and an odd vector trans-
lation are K-continuous maps (see Remark 3.2). In addition, a map preserving Khalimsky adjacency
(or a KA-map) does not allow a constant map because the KA-relation deals with only distinct points
(see (3.1)). To overcome this difficulty, the recent paper [13] develops two maps called an A-map
and an A-isomorphism which are expansions of a K-continuous map and a K-homeomorphism,
respectively.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8
1124 S.-E. HAN
To study several types of digital neighborhoods and their properties, we need to recall the digital
k-adjacency relation of Zn and a digital k-neighborhood. As a generalization of the k-adjacency
relations of 2D and 3D digital spaces [16, 19], the k-adjacency relations (or digital k-connectivity)
of Zn have been established [5] (see also [9, 10]).
For the natural numbers m,n with 1 ≤ m ≤ n, two distinct points p = (pi)i∈[1,n]Z and q =
= (qi)i∈[1,n]Z ∈ Zn are called k(m,n)- (for brevity, k-) adjacent if there are at most m indices i such
that |pi − qi| = 1 and for all other indices i, pi = qi.
Concretely, the k(m,n)-adjacency relations of Zn are determined according to the numbers
m,n ∈ N [5] (see also [9, 10]) as follows:
k := k(m,n) =
n−1∑
i=n−m
2n−iCni , where Cni =
n!
(n− i)! i!
. (2.1)
A pair (X, k) (or digital image) is assumed to be a (binary) set X ⊂ Zn with one of the k-
adjacency relations (see (2.1)) in a quadruple (Zn, k, k̄,X), where (k, k̄) ∈ {(k, 2n), (2n, 3n − 1)}
with k 6= k̄, k represents an adjacency relation for X and k̄ represents an adjacency relation for
Zn \X [16].
Using the adjacency relations of (2.1), in Zn we say that a digital k-neighborhood of p is
the set [19] Nk(p) := {q | p is k-adjacent to q}. Furthermore, we often use the notation [16]
N∗k (p) := Nk(p) ∪ {p}.
Given a k-adjacency relation of Zn, a simple k-path with l + 1 elements in Zn is assumed to
be an injective sequence (xi)i∈[0,l]Z ⊂ Zn such that xi and xj are k-adjacent if and only if either
j = i + 1 or i = j + 1 [16]. If x0 = x and xl = y, then the length of the simple k-path, denoted
by lk(x, y), is the number l. A simple closed k-curve with l elements in Zn, denoted by SCn,lk [5],
is the simple k-path (xi)i∈[0,l−1]Z , where xi and xj are k-adjacent if and only if j = i+ 1(mod l) or
i = j + 1(mod l) [16].
For a digital image (X, k) let us recall a digital k-neighborhood with radius 1 which is a
generalization of N∗k (p) [5].
Nk(x, 1) = N∗k (x) ∩X. (2.2)
3. Some properties of maps between digital topological spaces. To map every k0-connected
subset of (X, k0) into a k1-connected subset of (Y, k1), the paper [19] established the notion of digital
continuity. Motivated by this continuity, we represent the digital continuity of maps between digital
images, which can be efficiently used for studying spaces X ⊂ Zn, n ∈ N.
Proposition 3.1 [5, 8]. Let (X, k0) and (Y, k1) be digital images in Zn0 and Zn1 , respectively.
A function f : X → Y is (k0, k1)-continuous if and only if for every x ∈ X f(Nk0(x, 1)) ⊂
⊂ Nk1(f(x), 1).
Based on these concepts, let us consider a digital topological category, denoted by DTC, consisting
of two things [5]:
the set of (X, k) in Zn as objects;
the set of (k0, k1)-continuous maps as morphisms.
In DTC, in case n0 = n1 and k0 = k1 := k, we will particularly use the notation DTC(k).
Since the digital image (X, k) is considered to be a set X ⊂ Zn with one of the adjacency
relation of (2.1), we use the term a (k0, k1)-isomorphism as in [6] (see also [12]) rather than a
(k0, k1)-homeomorphism as in [2].
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8
EXISTENCE OF THE CATEGORY DTC2(k) WHICH IS EQUIVALENT TO THE GIVEN CATEGORY KAC2 1125
Definition 3.1 [12] (see also [6, 8]). For two digital images (X, k0) in Zn0 and (Y, k1) in Zn1 ,
a map h : X → Y is called a (k0, k1)-isomorphism if h is a (k0, k1)-continuous bijection and further,
h−1 : Y → X is (k1, k0)-continuous.
In Definition 3.1, in case n0 = n1 and k0 = k1, we call it a k0-isomorphism.
Let us now recall some properties of Khalimsky adjacency induced by (Zn, Tn).
Definition 3.2 [15]. In (Z2, T 2), we say that two distinct points x, y in Zn are Khalimsky
adjacent if y ∈ SN(x) or x ∈ SN(y), SN(q) means the smallest open set containing the point
q ∈ Zn, q ∈ {x, y}.
In Definition 3.2, we can extend the notion into the case (Zn, Tn) [13].
For a point p ∈ Zn the Khalimsky adjacency neighbor of p, denoted by A(p), is defined [13, 18]
as follows (for the case (Z2, T 2), see [15]):
A(p) = {q ∈ N3n−1(p) | {p, q} is connected under (Zn, Tn)}. (3.1)
For a point p := (
α︷ ︸︸ ︷
0, . . . , 0,
β︷ ︸︸ ︷
1, . . . , 1 ) := 〈α, β〉 ∈ Zn, the Khalimsky adjacency neighbor of p is
defined as follows:
A(p) := ([−1, 1]αZ × {1}β ∪ {0}α × [0, 2]βZ) \ ({0}α × {1}β). (3.2)
According to (3.2), since A(p) does not contain the point p := 〈α, β〉 , we obtain that cardinality of
A(p) is (3α − 1) + (3β − 1) = 3α + 3β − 2 [18].
Example 3.1 [13]. Let (Zn, Tn) be the Khalimsky nD space. Then we obtain the Khalimsky
adjacency neighbor of a point p := 〈α, β〉 ∈ Zn (for short, A(p) := A 〈α, β〉) as follows:
If n = 2, then for a point p ∈ Z2 we obtain A(p) = N8(p) if p is a pure point, and A(p) = N4(p)
if p is a mixed point. If n ≥ 3, then for a point p ∈ Zn A(p) = N3n−1(p) if p is a pure point, and if
given a point p := (pi)i∈[1,n]Z ∈ Zn is a mixed point, then according to the component of the given
coordinates pi, A(p) is determined by the method suggested in (3.2).
To investigate some properties of maps between K-topological spaces, for a space (X,TnX) := X
let us recall the Khalimsky adjacency relation for any two points in X as follows:
Definition 3.3 [18]. For (X,TnX) := X assume that AX(p) := A(p) ∩X. We say that for two
distinct points p, q ∈ X they are Khalimsky adjacent to each other if q ∈ AX(p) or p ∈ AX(q).
In view of Definition 3.3, we observe that Khalimsky adjacency holds only the symmetric relation
without the reflexive relation. In (Zn, Tn), since every point x ∈ Zn has its smallest neighborhood
SN(x), we say that a map f : (X,Tn0) := X → (Y, Tn1) := Y is K-continuous [13] if for every
point x ∈ X we have
f(SN(x)) ⊂ SN(f(x)). (3.3)
Besides, we say that the map f is aKA-map [13] if for two Khalimsky adjacent points x1, x2 ∈ X
the images f(x1), f(x2) are Khalimsky adjacent, which we define the following terminology [13].
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8
1126 S.-E. HAN
Definition 3.4. For a space (X,TnX) := X we define the following:
(1) Two distinct points x, y ∈ X are called Khalimsky adjacency (for brevity, A-) connected if
there is a sequence (or path) (xi)i∈[0,l]Z on X with {x0 = x, x1, . . . , xl = y} such that xi and xi+1
are Khalimsky adjacent, i ∈ [0, l − 1]Z, l ≥ 1. This sequence is called an A-path. Furthermore, the
number l is called the length of this A-path. Furthermore, an A-path is called a closed A-curve if
x0 = xl [13].
(2) A simple A-path in X is the sequence (xi)i∈[0,l]Z such that xi and xj are Khalimsky adjacent
if and only if either j = i+ 1 or i = j + 1.
Furthermore, we say that a simple closed A-curve with l elements (xi)i∈[0,l]Z is a simple A-path
with x0 = xl such that xi and xj are Khalimsky adjacent if and only if either j = i + 1(mod l) or
i = j + 1(mod l), l ≥ 4 [13].
Hereafter, we denote SCn,lA := (xi)i∈[0,l−1]Z by a simple closed A-curve with l elements in Zn,
n ∈ N− {1}, l ≥ 4 [13].
(3) A K-continuous map h : (X,Tn0
X )→ (Y, Tn1
Y ) is called a local K-homeomorphism if for any
x ∈ X, h maps SN(x) K-homeomorphically onto SN(h(x)) ⊂ Y.
In view of Definition 3.4(3), it is clear that while a K-homeomorphism implies a local K-
homeomorphism, the converse does not hold.
In (Zn, Tn) we say that a simple closed K-curve with l elements in Zn is a path (xi)i∈[0,l]Z ⊂ Zn
that is K-homeomorphic to a quotient space of a Khalimsky line interval [a, b]Z in terms of the
identification of the only two end points a and b. We denote it by SCn,lK := (xi)i∈[0,l−1]Z , l ≥ 4 [13].
Remark 3.1. In view of the property of SCn,lK , we say that it is a finite subspace in (Zn, Tn)
which is locally K-homeomorphic to the Khalimsky line (Z, T ).
Now we refer a difference between a K-continuous map and a KA-map.
Example 3.2. (1) Consider two spaces X, Y in Fig. 1(1). While X is considered to be both
SC2,4
K and SC2,4
A , Y is neither SC2,4
K nor SC2,4
A because Y cannot be connected under K-topology.
(2) Consider the following five spaces X, Y, Z, V and W in Fig. 1(2). While each of X, Y, V
and W is both SC2,8
K and SC2,8
A , Z is neither SC2,8
K (see the points z2, z6) nor SC2,8
A (see the points
zi, i ∈ {0, 2, 4, 6}).
(3) Consider the spaces X in Fig. 1(2) and D in Fig. 1(3). While these consist of eight elements,
the former contains mixed points and the latter contains only pure points. However, they have the
same structure as SC2,8
K as well as SC2,8
A .
Although both a K-continuous map and a KA-map play important roles in studying spaces
(X,TnX), they have some limitations as follows:
Remark 3.2 [13]. As said in the previous part, we observe that not every one click rotation of a
set X ⊂ Zn and an odd vector translation are K-continuous maps. To be specific,
(1) Consider the self-map f : SCn,lK := (ci)i∈[0,l−1] → SCn,lK given by f(ci) = ci+1(mod l). Then
f cannot be a K-continuous map (see the space X in Fig. 1(1)).
(2) Let us consider the map f : (Z, T )→ (Z, T ) given by f(t) = t+ (2n+ 1), n ∈ Z, which is
a parallel translation with an odd vector. Then we clearly observe that f cannot be a K-continuous
map.
(3) In addition, a KA-map also has the following limitation: a Khalimsky adjacency map does
not allow a constant map. When we consider a KA-relation of two points p and q in Zn, we always
assume that the given two points are distinct.
In view of Remark 3.2, we strongly need to develop another map overcoming the limitations (see
Definition 4.2). Besides, SCn,lK has the following property.
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EXISTENCE OF THE CATEGORY DTC2(k) WHICH IS EQUIVALENT TO THE GIVEN CATEGORY KAC2 1127
y
y
X
Y
y
y
Z
z
z
z
zz
z
z
z
V
d
4
5
d
w
w
w
6
d
3
7
d
d
d
10
7
6
3
w w w
w
5
b
b
b
1
0
3
b b
b
5
D
YX
1x
2x
3x
0x
4x
5x
6x
7x
3y
4y
5y
6y
7y
f
vv
v
v10
7
6
3
v v v
v
5
(1) (2) (3)
W
B
w2
d2
x2
x1x3
x0
d1
Fig. 1. Explanation of SCn,l
K and SCn,l
A .
Lemma 3.1. For SCn,lK := (xi)i∈[0,l−1]Z , we obtain that
(1) l 6= 6,
(2) l is an even number such that l ∈ 2N \ {6} and l ≥ 4.
Proof. (1) Let us take any point xi ∈ SCn,lK . Then, according to the structure of SCn,lK , we may
assume the following six cases:
(1) assume xi is pure closed and xi+1(mod l) is pure open;
(2) assume xi is pure closed and xi+1(mod l) is mixed;
(3) assume xi is pure open and xi+1(mod l) is pure closed;
(4) assume xi is pure open and xi+1(mod l) is mixed;
(5) assume xi is mixed and xi+1(mod l) is pure closed;
(6) assume xi is mixed and xi+1(mod l) is pure open.
Depending on the point xi ∈ SCn,lK above, each point xi has its smallest open neighborhood as
follows:
SN(xi) = (xi−1(mod l), xi, xi+1(mod l)) if xi is pure closed or mixed, and
SN(xi) = {xi} if xi is pure open.
To establish SCn,6K := (xi)i∈[0,5]Z , we may take x0 ∈ SCn,lK as a pure open, a pure closed or
a mixed point. Firstly, in case the given point x0 ∈ SCn,lK is a pure open point, we can consider
Case (3) or (4) above. Let us take (Case 3). Then the point x1 can be pure closed (see Fig. 2(1-1))
and further, according to (Case 2), we can take x2 as a mixed point. Then we fail to make SCn,6K . If
we take x2 as an pure open point via (Case 1), then we cannot have SCn,6K either.
As the other case, let us follow (Case 4) to take point x1 as a mixed point (see Fig. 2(1-2)). Then
we should take x2 as a pure open point in N8(x1) (see Cases (5), (6)), which we fail to have SCn,6K .
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1128 S.-E. HAN
Secondly, in case the given point x0 ∈ SCn,lK is a pure closed point, the proof is completed by
using the method similar to the first case (see Fig. 2(2-1) and (2-2)).
Finally, in case the given point x0 ∈ SCn,lK is a mixed point, the proof is also completed by using
the method similar to the first and second cases (see Fig. 2(3-1) and (3-2)).
(2) According to the properties of Cases (1) – (6) above, to establish SCn,lK := (xi)i∈[0,l−1]Z ,
when starting at a point x0 ∈ Zn as an pure open, pure closed or mixed point, we cannot have the
consecutive points xi, xi+1 (mod l) such that they are all pure open, pure closed or mixed points. Thus,
by the properties of Cases (1) – (6) above, the number l of SCn,lK should be even. By the property of
(1) of this theorem, we obtain l 6= 6. However, we have SCn,4K (see Fig. 1(1)).
Lemma 3.1 is proved.
(3-1)(2-1)(1-1) (3-2)(1-2) (2-2)
X0
X1
X2 X1
X2
X0
X2
X1
X0 X0X1
X1 X0X0
X1
X2
Fig. 2. Several cases of the points related to SC2,6
K .
Theorem 3.1. Consider two spaces SCn1,l1
K and SCn2,l2
K . They are K-homeomorphic to each
other if and only if l0 = l1.
Proof. Owing to the topological structure of SCn,lK := (xi)i∈[0,l−1]Z , for i ∈ [0, l − 1]Z the
subspace {xi, xi+1(mod l)} is a connected set and we need to recall that the number l should be
even (see Lemma 3.1). If not, it cannot be locally K-homeomorphic to (Z, T ). Besides, each point
xi ∈ SCn,lK , i ∈ [0, l − 1]Z, has the smallest open neighborhood satisfying
in case SN(xi) = {xi−1(mod l), xi, xi+1(mod l)},
we obtain SN(xi+1(mod l)) = {xi+1(mod l)};
and in case SN(xi) = {xi},
we have SN(xi+1(mod l)) = {xi(mod l), xi+1(mod l), xi+2(mod l)}.
(3.4)
Consequently, for two spaces SCn1,l1
K := (xi)i∈[0,l1−1]Z and SCn2,l2
K := (yj)j∈[0,l2−1]Z if l1 = l2,
then we have a map f : SCn1,l1
K → SCn2,l2
K satisfying that the restriction map on SN(xi), denoted
by f |SN(xi), is (locally) K-homeomorphic to SN(yj) for each i ∈ [0, l1 − 1]Z. Thus the map f is a
K-homeomorphism.
Conversely, if SCn1,l1
K is K-homeomorphic to SCn2,l2
K , then it is clearly l1 = l2.
Theorem 3.1 is proved.
In relation to the geometric transformation of SCn,lK , we obtain the following property of a rotation
of SCn,lK . In view of the structure of SCn,lK and Lemma 3.1, since the number l should be even, we
obtain the following theorem.
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Theorem 3.2. Let h : SCn,2lK := (xi)i∈[0,2l−1]Z → SCn,2lK be the self-map given by h(xi) =
= xi+l(mod 2l), l ∈ N \ {1, 3}. Then h is a K-homeomorphism.
Proof. For a given SCn,2lK , the point xi ∈ SCn,2lK has the property (3.4). Furthermore, by
Lemma 3.1 and further, according to the Cases (1) – (6) in the proof of Lemma 3.1, we can define the
map h : SCn,2lK → SCn,2lK given by h(xi) = xi+l(mod 2l), which is a K-homeomorphism.
Theorem 3.2 is proved.
4. Establishment of two categories of K-topological spaces with Khalimsky adjacency. To
overcome the limitations discussed in Remark 3.2, in this section we formulate two categories
associated with K-topology which can be substantially helpful to study a space (X,TnX). In relation
to the establishment, we will use the following Khalimsky adjacency neighborhood of a point p ∈ X.
Definition 4.1 [13]. For a space (X,TnX) := X and a point p ∈ X we define a Khalimsky
adjacency neighborhood of p to be the set AX(p) ∪ {p} := ANX(p).
Hereafter, in (X,TnX) we will for brevity use AN(p) instead of ANX(p) if there is no danger of
ambiguity. Indeed, using AN(p), we develop an A-map and an A-isomorphism (see Definitions 4.2
and 4.3).
Definition 4.2 [13]. For two spaces (X,Tn0
X ) := X and (Y, Tn1
Y ) := Y, we say that a function f :
X → Y is an A-map at a point x ∈ X if
f(AN(x)) ⊂ AN(f(x)).
Furthermore, we say that a map f : X → Y is an A-map if the map f is an A-map at every point
x ∈ X.
Indeed, an A-map can be also represented by using a smallest neighborhood [13]:
f : (X,Tn0
X ) := X → (Y, Tn1
Y ) := Y is an A-map
if for every x, x′ in X such that x′ ∈ SN(x) or x ∈ SN(x′)
it holds that f(x′) ∈ SN(f(x)) or f(x) ∈ SN(f(x′)).
(4.1)
According to (4.1), we observe that an A-map implies a map preserving connected subsets of X into
connected ones and further, it generalizes both aK-continuous map and aKA-map (see Example 4.1).
Thus an A-map can be useful for studying K-topological spaces with K-adjacency.
Example 4.1. Consider the map f : X → Y in Fig. 1(2) given by f(xi) = yi, i ∈ [0, 7]Z. While
the map f cannot be a K-continuous map, it is an A-map.
Using spaces (X,TnX) := X and A-maps, we establish a category of K-topological spaces with
Khalimsky adjacency denoted by KAC in terms of the following two sets [13].
(1) The set of spaces (X,TnX) := X with K-adjacency as objects denoted by Ob(KAC );
(2) The set of A-maps between all pairs of elements in Ob(KAC ) as morphisms.
Since (Zn, Tn) is an Alexandroff T0-space, we obtain that in (Zn, Tn), for two distinct points
p and q in Zn the point p is Khalimsky adjacent to q if and only if the set {p, q} is connected
[13, 18], in case (Z2, T 2) the property was studied in the paper [15]. Thus, in view of (3.1), (3.2)
and Definition 4.1, since for each point p ∈ Zn SN(p) ⊂ AN(p), we obtain the following corollary.
Corollary 4.1. For a point p ∈ X := (X,TnX) we obtain SN(p) ⊂ AN(p) ⊂ N3n−1(p, 1).
By Example 4.1, Corollary 4.1 and Definition 4.2, we obtain the following theorem.
Theorem 4.1 [13]. Let f : (X,Tn0
X ) := X → (Y, Tn1
Y ) := Y be a map.
Every K-continuous map is an A-map. But the converse does not hold.
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1130 S.-E. HAN
Definition 4.3 [13]. For two spaces (X,Tn0
X ) := X and (Y, Tn1
Y ) := Y in KAC, a map h :
X → Y is called an A-isomorphism if h is a bijective A-map (for short, A-bijection) and if h−1 :
Y → X is an A-map.
Definition 4.4. We say that an A-map h : X → Y is a local A-isomorphism if for any x ∈ X,
h maps AN(x) A-isomorphically onto AN(h(x)) ⊂ Y.
In view of Definitions 4.3 and 4.4, it is clear that while an A-isomorphism implies a local
A-isomorphism, the converse does not hold.
Using the method similar to the establishment of SCn,lK ,we can rewrite SCn,lA in such a way: SCn,lA
is a finite space in Zn that is locally A-isomorphic to the Khalimsky line with Khalimsky adjacency,
l ≥ 4 because in (Z, T ) we obtain that for each point p ∈ Z AN(p) = {p − 1, p, p + 1}. However,
SCn,lA and SCn,lK have different structures [13]. While each point ci ∈ SCn,lK := (ci)i∈[0,l−1]Z
has either SN(ci) = {ci} or SN(ci) = {ci−1(mod l), ci, ci+1(mod l)}, each point xi ∈ SCn,lA :=
:= (xi)i∈[0,l−1]Z has AN(xi) = {xi−1(mod l), xi, xi+1(mod l)}.
Example 4.2. Consider the spaces B and D in Fig. 1(3). While D is a kind of SC2,8
A , B is
neither SC2,6
A nor SC2,6
K (see the point b3 and further, Lemma 3.1).
5. Development of an A-homotopy in KAC1 and an A-homotopy equivalence. In KAC1
the paper proposes the notions of an A-homotopy and an A-homotopy equivalence, and classifies
spaces in KAC1 or KAC2 in terms of an A-homotopy equivalence. Finally, the paper proves that
KAC2 is equivalent to DTC2(k) for any k-adjacency of Zn in (2.1). For a space X in KAC1,
let B be a subset of X in KAC1. Then (X,B) is called a space pair in KAC1. Furthermore, if B
is a singleton set {x0}, then (X,x0) is called a pointed space in KAC1. Motivated by the pointed
digital homotopy in [2] and the digital relative homotopy in [7], we will establish the notions of an
A-homotopy relative to a subset B ⊂ X, an A-homotopy equivalence, which will be helpful to study
spaces in KAC1.
Definition 5.1. Let (X,B) and Y be a space pair and a space in KAC1, respectively. Let f, g :
X → Y be A-maps. Suppose there exist m ∈ N 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 given by Fx(t) = F (x, t) for all
t ∈ [0,m]Z is an A-map;
for all t ∈ [0,m]Z, the induced function Ft : X → Y given by Ft(x) = F (x, t) for all x ∈ X
is an A-map.
Then we say that F is an A-homotopy between f and g.
Furthermore, for all t ∈ [0,m]Z, assume that the induced map Ft on B is a constant which
follows the prescribed function from B to Y. In other words, Ft(x) = f(x) = g(x) for all x ∈ B and
for all t ∈ [0,m]Z.
Then we call F an A-homotopy relative to B between f and g, and we say that f and g are
A-homotopic relative to B in Y, f 'A relB g in symbols.
In Definition 5.1, if B = {x0} ⊂ X, then we say that F is a pointed A-homotopy at {x0}. In
case f and g are pointed A-homotopic in Y, we use the notation f 'A g. In addition, if n0 = n1,
then we say that f and g are pointed A-homotopic in Y and we use the notation that f 'A g and
f ∈ [g] which denotes the A-homotopy class of g. If, for some x0 ∈ X, 1X is A-homotopic to the
constant map in the space x0 relative to {x0}, then we say that (X,x0) pointed A-contractible.
Motivated by the notion of a digital homotopy equivalence [4], we propose the following defini-
tion.
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Definition 5.2. For two spaces (X,Tn1
X ) and (Y, Tn2
Y ) in KAC1, if there is an A-map h :
X → Y and an A-map l : Y → X such that l ◦ h is A-homotopic to 1X and h ◦ l is A-homotopic to
1Y , then the map h : X → Y is called an A-homotopy equivalence and denote it by X 'A·h·e Y.
Example 5.1. Consider the spacesX and Y inKAC1 (see Fig. 3). Using the processes presented
by the arrows in Fig. 3, we observe that they are A-homotopy equivalent to each other because the
spaces X ′ and Y ′ in Fig. 3 are A-isomorphic to each other.
(0, 0, 0)
0b
3
b
b
b
5
X
b
b b
1
2
4
7 b6
0c
1
c
c
4
c
c c
2 3
5
c 6c
7
Y
(0, 0, 0)
0b
3
b
b
b
5
X
b
b b
1
2
4
7 b6
0c
1
c
c
4
c
c c
2 3
5
c 6c
7
Y
Fig. 3. Explanation of an A-homotopy equivalence.
Let us recall the category KAC2 whose objects are simple closed A-curves and morphisms are
A-maps so that KAC2 is a subcategory of KAC1. In view of the property of SCn,lA , we obtain the
following theorem.
Theorem 5.1. In KAC2, consider two spaces SCni,li
A , i ∈ {1, 2}. They are A-homotopy equiv-
alent to each other if and only if l1 = l2.
Proof. Firstly, in case all SCni,li
A , i ∈ {1, 2}, are A-contractible, we easily see that l1 = 4 = l2.
Secondly, one of these is A-contractible. Consider SCn1,l1
A with l1 = 4 and SCn2,l2
A which is not
A-contractible. Then l2 is greater than 4, the proof is completed.
Finally, as the other cases, without loss of generality, assume l1 � l2 with l1
4. Let f :
SCn1,l1
A → SCn2,l2
A be any A-map. From the hypothesis l1 � l2, it follows that f(SCn1,l1
A ) is a proper
and A-connected subset of SCn2,l2
A . Then f(SCn1,l1
A ) is A-contractible in SCn2,l2
A . It implies that if g :
SCn2,l2
A → SCn1,l1
A is any A-map, then g ◦ f(SCn1,l1
A ) is A-contractible in SCn1,l1
A . Since SCn2,l2
A
is not A-contractible, 1
SC
n1,l1
A
and g ◦ f cannot be A-homotopic in SCn1,l1
A .
Conversely, if l1 = l2, then SCn1,l1
A is A-isomorphic to SCn2,l2
A . Hence we obtain SCn1,l1
A 'A·h·e
'A·h·e SCn2,l2
A .
Theorem 5.1 is proved.
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1132 S.-E. HAN
6. Existence of the category DTC2(k) which is equivalent to the given category KAC2.
This section proves that given the category KAC2 there is the category DTC2(k) which is equivalent
to KAC2, so that for any SCn,lA we prove that there is SCn,lk which is equivalent to SCn,lA , where
the k-adjacency is that of the digital connectivity of Zn in (2.1). Unlike Remark 3.2(1), we have the
following property supporting rotations of the space SCn,lA under an A-isomorphism.
Theorem 6.1. Let f : SCn,lA := (xi)i∈[0,l−1]Z → SCn,lA be the self-map given by f(xi) =
= xi+m(mod l),m ∈ N. Then f is an A-isomorphism.
Proof. Let f : SCn,lA → SCn,lA be the map given by f(xi) = xi+m(mod l),m ∈ N. Since each
point xi ∈ SCn,lA has AN(xi) = {xi−1(mod l), xi, xi+1(mod l)}, the map f is an A-bijection and its
inverse f−1 is also an A-map.
Theorem 6.1 is proved.
Motivated by Theorem 6.1, we obtain the following (indeed, a small part of Theorem 6.4 in [13]
was missing):
Theorem 6.2 (correcting Theorem 6.4 in [13]). SCn,l1A is A-isomorphic to SCn,l2A if and only if
l1 = l2.
Although two spaces SCn,lA and SCn,lk have different features because the former is considered
in KAC2 and the latter is considered in DTC2(k), they are equivalent to each other.
Theorem 6.3. For any SCn,lA ∈ Ob(KAC2), there is both SCn,lk ∈ Ob(DTC2(k)) and a
bijection h : SCn,lA → SCn,lk preserving the KA-relation of SCn,lA into the k-connectivity of SCn,lk
and its inverse h−1 : SCn,lk → SCn,lA preserving the k-connectivity of SCn,lk into the KA-relation of
SCn,lA , where the k-adjacency is any k-adjacency relation of Zn in (2.1).
Proof. Consider SCn,lA := (ci)i∈[0,l−1]Z and put SCn,lk := (di)i∈[0,l−1]Z . Owing to the properties
of SCn,lA and SCn,lk , each point ci ∈ SCn,lA has an AN(ci) = {ci−1(mod l), ci, ci+1(mod l)} and further,
each point di ∈ SCn,lk hasNk(di, 1) = {di−1(mod l), di, di+1(mod l)} ⊂ SC
n,l
k . Then, define the map h :
SCn,lA → SCn,lk given by h(ci) = di+m(mod l), m ∈ N so that it is a bijection. Besides, if each ci,
i ∈ [0, l−1]Z, is Khalimsky adjacent to ci+1(mod l), then the images by the map h are also k-adjacent
in SCn,lk . This implies that the map h preserves a KA-relation of SCn,lA into a k-adjacent relation
of SCn,lk like a homomorphism in category theory. Similarly, the inverse of the map h, denoted by
h−1, preserves a k-adjacent relation of SCn,lk into a KA-relation of SCn,lA . Besides, the compositions
h ◦ h−1 and h−1 ◦ h are obviously identities on SCn,lk and SCn,lA , respectively.
Theorem 6.3 is proved.
Theorem 6.4. Given the category KAC2, there is a category DTC2(k) which is equivalent to
KAC2.
Proof. (1) Objects: In general, for two categories C and D, an equivalence between C and
D requires that the space X ∈ Ob(C) is equivalent to Y ∈ Ob(D). Namely, there are equivalent
functors F : C → D and G : D → C between C and D, and two natural isomorphisms G ◦ F = 1F
and F ◦G = 1G. Thus we obtain that F |Ob(C) : Ob(C)→ Ob(D), G|Ob(D) : Ob(D)→ Ob(C), and
F |Ob(C)(X) = Y, G|Ob(D)(Y ) = X.
To be specific, consider the map F : KAC2 → DTC2(k) such that for any SCn,lA ∈ Ob(KAC2)
F (SCn,lA ) = SCn,lk and for any f ∈ Mor(KAC2) F (f) = f. Thus for any object X ∈ KAC2 and
each point x ∈ X := SCn,lA , we have AN(x) which is equivalent to Nk(x, 1) ⊂ SCn,lk ∈ DTC2(k).
By Theorem 6.3, we obtain that
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EXISTENCE OF THE CATEGORY DTC2(k) WHICH IS EQUIVALENT TO THE GIVEN CATEGORY KAC2 1133
F |Ob(KAC2) : Ob(KAC2)→ Ob(DTC2(k)), G|Ob(DTC2(k)) : Ob(DTC2(k))→ Ob(KAC2),
such that
F |Ob(KAC2)(SC
n,l
A ) = SCn,lk , (F |Ob(KAC2))
−1 := G|Ob(DTC2(k))(SC
n,l
k ) = SCn,lA .
(2) Morphisms: Owing to their own structures of AN(x) and Nk(x, 1), an A-map is equivalently
considered to be a k-continuous map in DTC2(k) (see Proposition 3.1 and Definition 4.2).
(3) Since the morphisms in KAC2 and DTC2(k) have the transitive property, we consider the
following two functors F : KAC2 → DTC2(k) satisfying F (f1 ◦1 f2) = F (f1) ◦2 F (f2) and G :
DTC2(k)→ KAC2 satisfying G(g1 ◦2 g2) = G(g1)◦1G(g2), where “◦1” and “◦2” are compositions
in the morphisms ofKAC1 andDTC2(k), respectively. Then we obtain two natural isomorphisms ε :
F ◦G → IG and η : IF → G ◦ F, where IF and IG are identity functors on KAC2 and DTC2(k),
respectively.
Theorem 6.4 is proved.
Example 6.1. Let us consider SCn,lA ∈ KAC2 such as SC2,8
A , to be specific, the spaces X, Y,
V and W in Fig. 1(2). Then we obtain SCn,lk ∈ DTC2(k) such as SC2,8
4 and SC2,8
8 such that both
X and Y (resp. V and W ) are equivalent to SC2,8
8 (resp. SC2,8
4 ).
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Received 04.10.12,
after revision — 19.05.15
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8
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