Semi-Jordan curve theorem on the Marcus-Wyse topological plane

: The paper initially develops the semi-Jordan curve theorem on the digital plane with the Marcus-Wyse topology, i.e., MW -topological plane or ( Z 2 , γ ) for brevity. We ﬁrst prove that while every simple closed MW -curve is semi-open in ( Z 2 , γ ), it may not be semi-closed. Given a simple closed MW -curve with l elements, denoted by S C l γ , after establishing a continuous analog of S C l γ denoted by A ( S C l γ ), we initially show that A ( S C l γ ) is both semi-open and semi-closed in ( R 2 , U ), where ( R 2 , U ) is the 2-dimensional real plane R 2 with the usual topology U . Furthermore, we ﬁnd a condition for A ( S C l γ ) to separate ( R 2 , U ) into exactly two non-empty components, compared to a typical Jordan curve theorem on ( R 2 , U ). Since not every S C l γ always separates ( Z 2 , γ ) into two nonempty components, we ﬁnd a condition for S C l γ , l (cid:44) 4 , to separate ( Z 2 , γ ) into exactly two components. The semi-Jordan curve theorem on the MW -topological plane plays an important role in applied topology such as digital topology, mathematical morphology as well as computer science.


Introduction
In this paper, we use the notation Z (resp. N and R) to indicate the set of integers (resp. natural numbers and real numbers). Besides, since we will often use the name "Marcus-Wyse" in this paper, we will take the term "MW-" instead of "Marcus-Wyse", if there is no danger of ambiguity. Besides, Z o (resp. N e ) means the set of odd integers (resp. even natural numbers) and further, we will use the notation "⊂" (resp. X ) to denote a 'proper subset or equal' (resp. the cardinality of the given set X). The notation " :=" will be used to introduce a new notion or a terminology. In addition, let us denote a simple closed MW-curve with l elements by S C l γ , 4 ≤ l ∈ N e \ {6} (see Definition 2.1(3) in detail).
Indeed, the well-known Jordan curve theorem on the 2-dimensional real space [1] has some limitations of dealing with digital objects on Z 2 from the viewpoints of applied sciences such as digital topology and digital geometry. Thus, in relation to the establishment of several types of Jordan curve theorems in digital topological settings, there are many works including the papers [2][3][4][5][6][7][8][9][10][11][12]. In the literature, to do this work, digital graph theory [7][8][9] and several types of topologies have been used such as Khalimsky, Marcus-Wyse, Alexandroff topology, pretopology, and so on. However, given a certain topological space (Z 2 , T ), the earlier works did not examine topological features of J and Z 2 \ J, where J is a simple closed digital curve in (Z 2 , T ). Since both J and Z 2 \ J may not be either a closed or an open set in (Z 2 , T ), we need to intensively study some topological features of both J and Z 2 \ J. Furthermore, with a certain topological space (Z 2 , T ), since the number of the components of Z 2 \ J can be very important from the viewpoint of mathematics, we need to intensively investigate this topic. For instance, on the MW-topological plane, i.e., (Z 2 , γ), the present paper clearly shows that the number of the components of the complement of S C l γ in (Z 2 , γ) depends on the situation. Besides, we also find that topological features of the sets S C l γ and Z 2 \ S C l γ are so related to the semi-closedness and semi-openness in (Z 2 , γ). In detail, see [13,14] or Section 3 in the present paper. Indeed, there are lots of works studying various properties of semi-closed and semi-open subsets of a topological space [13][14][15][16][17][18][19]. Based on this approach, the present paper will partially use these works.
The aim of the present paper is initially to propose the semi-Jordan curve theorem on the digital plane with the MW-topology (or (Z 2 , γ)) because it has something quite independent from the earlier results in the literature including the papers [3,4,6,8,9,11,12]. To propose this theorem and support some utilities, we will mainly deal with the following topics.
• Examination of many types of S C l γ with respect to the semi-closedness and semi-openness in (Z 2 , γ). • Establishment of a method for making a continuous analog of S C l γ and an investigation of some topological features of A(S C l γ ) with respect to the semi-openness and semi-closedness in (R 2 , U), where (R 2 , U) is the 2-dimensional real space with the usual topology.
• Given an S C l γ , l 4, how to separate (R 2 , U) in terms of A(S C l γ ) ? • Assume the two subspaces (X, γ X ) and (Y, γ Y ) that are MW-homeomorphic to S C l γ . Then we will examine if the number of the components of R 2 \ A(X) is equal to that of R 2 \ A(Y). Besides, we strongly need to further compare the number of the components of Z 2 \ X and that of Z 2 \ Y.
• Given an S C l γ , we need to examine if the number of the components of Z 2 \ S C l γ is a topological invariant.
• Under what condition, does S C l γ separate (Z 2 , γ) into exactly two components? • Development of the semi-Jordan curve theorem on the MW-topological plane. Besides, given an S C l γ , we investigate how to separate (Z 2 , γ) with respect to the semi-Jordan curve theorem. • Investigation of some properties of many kinds of S C l γ relating to the semi-Jordan curve theorem. After addressing these topics with a success, we can confirm that the semi-Jordan curve theorem has strong advantages and some utilities compared with the earlier works in the literature because it does not have any paradox raised in the Rosenfeld's approach and further, it proceeds with the topological structures, which makes a distinction from the Rosenfeld's approach.
This paper is organized as follows: Section 2 provides some basic notions related to the digital k-connectivity on Z 2 and the MW-topology. Section 3 studies some tools discriminating between semi-open and semi-closed sets in (Z 2 , γ) and further, investigates various properties of semi-closed or semi-open subsets in (Z 2 , γ). In Section 4, after examining if a simple closed MW-curve is semi-open and semi-closed in (Z 2 , γ), we prove the semi-openness of each S C l γ and further, the semi-closedness of it is related to the number of l. Section 5 suggests a method for establishing a continuous analog of an S C l γ denoted by A(S C l γ ) by using the local rule in [20,21] considered on R 2 . Besides, we find some conditions for A(S C l γ ) to separate (R 2 , U) into exactly two components, compared to the typical Jordan curve theorem in (R 2 , U) that is the 2-dimensional real plane. Furthermore, we prove that A(S C l γ ) is both a semi-open and a semi-closed subset of (R 2 , U). Meanwhile, every semi-closed S C l γ , l 4, is proved to separate (Z 2 , γ) into many semi-open components whose number depends on the number l of S C l γ . Section 6 proposes the semi-Jordan curve theorem on the MW-topological plane. Besides, a semi-open S C l γ is also proved to separate (Z 2 , γ) into semi-closed or semi-open components whose number depends on the situation. More precisely, after proving that S C l γ separates (Z 2 , γ) into many semi-closed or semi-open components depending on the situation, we find a condition for S C l γ to separate (Z 2 , γ) into exactly two components. Besides, given two simple closed MW-curve with l elements X and Y, we first prove that the number of components of X c need not be equal to that of Y c . Section 7 refers to some advantages and utilities of MW-topological structure and the semi-Jordan curve theorem on (Z 2 , γ). Section 8 concludes the paper with summary and a further work.

Preliminaries
To study digital objects in Z 2 , many basic notations will be used such as a digital 4-and 8-neighborhood of a point p ∈ Z 2 [7,8], as follows: Based on the digital 4-and 8-connectivity in [7,8,22], for a point p = (x, y) ∈ Z 2 , the following notations will be often used later [7,8].
We now recall an Alexandroff topological structure using the study of some properties of MWtopological spaces. More precisely, a topological space (X, T ) is called an Alexandroff space if every point x ∈ X has the smallest open neighborhood in (X, T ) [24]. As an Alexandroff topological space [24,25], the Marcus-Wyse topological space, denoted by (Z 2 , γ), was established and there are many studies including the papers [5,6,26]. Indeed, the MW-topology, denoted by (Z 2 , γ), is generated by the set of all U(p) in (2.2) below, i.e., {U(p) | p ∈ Z 2 }, as a base [27], where for each point In the paper we call a point p = (x 1 , x 2 ) doubly even if x 1 + x 2 is an even number such that each x i is even, i ∈ {1, 2}; even if x 1 + x 2 is an even number such that each x i is odd, i ∈ {1, 2}; and odd if x 1 + x 2 is an odd number [12].
In all subspaces of (Z 2 , γ) of Figures 1-7 the symbols ♦ and • mean a doubly even point or even point and an odd point, respectively. In view of (2.2), we can obviously obtain the following: Under (Z 2 , γ), the singleton {♦} is a closed set and {•} is an open set. Besides, for a subset X ⊂ Z 2 , the subspace induced by (Z 2 , γ) is obtained, denoted by (X, γ X ) and called an MW-topological space. Hereinafter, for our purpose, we will use the notations In terms of this perspective, it turns out that the minimal (open) neighborhood of the point p := (p 1 , p 2 ) of Z 2 , denoted by S N γ (p) ⊂ Z 2 , is obtained, as follows [26,28]: Hereinafter, in (X, γ X ), for p ∈ X we use the notation S N γ (p) := S N γ (p) ∩ X for short if there is no danger of ambiguity. Using the smallest open set of (2.4), the notion of an MW-adjacency in (Z 2 , γ) is defined, as follows: For distinct points p, q in (Z 2 , γ), we say that p is MW-adjacent to q [26] if In view of the properties of (2.2) and (2.4), we obviously obtain the following: Based on the structure of (2.4), for a point p := (p 1 , p 2 ) of Z 2 , the closure of the singleton {p} is denoted by Cl γ {p}) ⊂ Z 2 as follows [26]: Hereinafter, in relation to the study of MW-topological spaces, we will use the term Cl for brevity instead of Cl γ if there is no danger of confusion.
Definition 2.1. [26] Let X := (X, γ X ) be an MW-topological space. Then we define the following: (1) An MW-path from x to y in X is defined as a sequence (p i ) i∈[0,l] Z ⊂ X, l ∈ N, in X such that p 0 = x, p l = y and each point p i is MW-adjacent to p i+1 and i ∈ [0, l − 1] Z . The number l is the length of this path. In particular, a singleton in (Z 2 , γ) is assumed to be an MW-path.
(2) Distinct points x, y ∈ X are called MW-path connected if there is a finite MW-path (p i ) i∈[0,m] Z on X with p 0 = x and p m = y. For arbitrary points x, y ∈ X, if there is an MW-path (p i ) i∈[0,m] Z ⊂ X such that p 0 = x and p m = y, then we say that X is MW-path connected (or MW-connected).
(3) A simple closed MW-curve (resp. simple MW-path) with l elements in X means a finite MW-path in Z 2 such that the points p i and p j are MW-adjacent if and only if | i − j | = ±1(mod l) (resp. | i − j | = 1). Then we use the notation S C l γ to denote a simple closed MW-curve with l elements.
As for some properties of S C l γ , it is clear that S C l 1 γ is MW-homeomorphic to S C l 2 γ if and only if l 1 = l 2 [26].    Let us further establish some techniques to examine if a set in (Z 2 , γ) is semi-open or semi-closed. In (Z 2 , γ), for a set X ⊂ Z 2 , we will take the following notation [31].
(2) If X is an open set in (Z 2 , γ), then there is at least an odd point x in X (see the property of (2.3)).
Given a set X in (Z 2 , γ), to further examine if the set X is semi-open or semi-closed in (Z 2 , γ), we now introduce the following two theorems that will be strongly used in discriminating against subsets based on the semi-openness and semi-closedness of the MW-topological space.
Since this theorem strongly plays an important role in studying many results in the present paper, to make Theorem 3.3 self-contained, we suggest a proof briefly, as follows: In case X = ∅, the proof is straightforward. Let us assume that X is not an empty set. (⇒) According to the choice of a point x ∈ X, we can consider the following two cases. (Case 1) Assume that x(∈ X) is an odd point. From the hypothesis, we have x ∈ X ⊂ Cl(Int(X)) so that we obtain S N γ (x) ∩ Int(X) ∅. (3.2) Since S N γ (x) = {x}, we obtain x ∈ Int(X) and further, x ∈ X op . Hence, owing to (3.2), we have (Case 2) Assume that x ∈ X is a doubly even or even point. Owing to the hypothesis, we obtain x ∈ Cl(Int(X)) that leads to the following property as mentioned in (3.2).
By the property of (3.3), since z ∈ Int(X) ⊂ X, we have z ∈ X op (see Remark 3.2(2)) so that z ∈ S N γ (x) ∩ X op ∅. In addition, we see that the point z is indeed MW-adjacent to x.
(⇐) According to the choice of a point x ∈ X, we can consider the following two cases. (Case 1) For an arbitrary point x ∈ X, assume that x is an odd point in (Z 2 , γ).
For an arbitrary point x ∈ X, assume that x is a doubly even or even point in (Z 2 , γ). Owing to the hypothesis, since S N γ (x) ∩ X op ∅, by Remark 3.2 (2) and (3), there is an odd point z in ( Owing to both (3.4) and (3.5), we obtain X ⊂ Cl(Int(X)) which prove the assertion.
Owing to the notion of semi-closedness, using Theorem 3.4, we obtain the following:

As examples for Theorems 3.3 and 3.4, see the cases referred to in Remark 3.1(1)-(3). In view of Theorems 3.3 and 3.4, we have the following:
Remark 3.5. In (Z 2 , γ), assume a connected subset X with X ≥ 2. Then it is semi-open and it may not be semi-closed.

Classification of simple closed MW-curves with respect to the semi-closedness
To classify all types of S C l γ with respect to the semi-openness and semi-closedness, based on the topological features of S C l γ , it suffices to consider the only case of l ∈ {2m | m ∈ N \ {1, 3}} because no S C 6 γ exists. Hereinafter, when studying semi-topological features of a set X ⊂ Z 2 , we assume that the set X is considered in (Z 2 , γ).
Theorem 4.1. Given an S C l γ , 4 ≤ l ∈ N e \ {6}, the semi-topological features of S C l γ in (Z 2 , γ) are determined according to the number l, as follows: (2) (2-1) In the case of S C 4 γ , let Y := Z 2 \ S C 4 γ . Then, for any p ∈ Y we have . Hence S C 4 γ is semi-closed in (Z 2 , γ). Also, using a method similar to the proof of (1), it is clear that S C 4 γ is semi-open in (Z 2 , γ).
(2-2) In the case of S C 10 γ (see the objects in Figure 1(3)(a),(b)), let W := Z 2 \ S C 10 γ . Using a method similar to the proof of (2-1) above, by Theorems 3.3 and 3.4, we prove that S C 10 γ is both semi-open and semi-closed in (Z 2 , γ). Proof: In the case of S C l γ , where l {4, 10}, the semi-closedness of S C l γ depends on the situation. More precisely, given an S C l γ : whose each element is an odd point (i.e., Then, by Theorem 3.4, S C l γ is semi-closed. For instance, since no S C 6 γ exists, it suffices to mention that S C l γ , l {4, 10}, is semi-closed depending on the situation. As suggested in Figure 1(2), while the object S C 8 γ of (a) is not semi-closed (see Theorem 3.4) but semi-open, the object of (b) is both semi-closed and semi-open.
In view of Theorem 4.1, we obtain the following: There are two types of S C l γ , l {4, 10}, with respect to the semi-closedness. Proof: Using Theorems 3.3 and 3.4, we prove the assertion. As mentioned in the proof of Theorem 4.1, we need to consider the following two cases: (Case 1) Assume an S C l γ := (x i ) i∈[0,l−1] Z , l {4, 10}, such that S C l γ does not have the subsequence whose each element is an odd point and , in which there is the subsequence X 1 of (4.1) whose each element is an odd point and X 1 ⊂ N 4 (x), x ∈ Z 2 \ S C l γ . Then, by Theorem 3.4, S C l γ is not semi-closed but only semi-open owing to Theorem 3.3. For instance, since S C 12 γ in Figure  2(a) does not satisfy the condition of Theorem 3.4, it is not semi-closed in (Z 2 , γ) (see the points γ as in Figure 2(b)).  Figure 2(b), owing to the two points p 1 , p 2 in Z 2 \ S C 12 γ in Figure 2(a), we conclude that S C 12 γ is not semi-closed in (Z 2 , γ) because Int(Cl(S C 12 γ )) S C 12 γ (see the points p 1 and p 2 in (b)). However, we obtain S C 12 γ ⊂ Cl(Int(S C 12 γ )) that implies the semi-openness of S C 12 γ .
satisfies only the condition of Theorem 3.3 instead of that of Theorem 3.4. To be specific, based on an S C 12 γ in Figure 2(a), consider the object in Figure 2

Establishment of a continuous analog of
This section introduces a method for establishing a continuous analog of an object on Z 2 with respect to the MW-topology. The local rule introduced in Definition 5.1 below will be used in this work and has been widely used in digitization and digital-based rough set theory [20].
Definition 5.1. [20] For each point p := (p 1 , p 2 ) ∈ Z 2 , the continuous analog of the given point p ∈ Z 2 with respect to the MW-topology, denoted by A p , is defined by: if p is a doubly even point (see Figure 3(1)); if p is an even point (see Figure 3(2)); and if p is an odd point (see Figure 3(3)-(4)).
Hereinafter, we assume the set A p to be a subspace of (R 2 , U). Using the local rule around a point p ∈ Z 2 as in Definition 5.1, we define the following: Figure 3. Configurations of A p (⊂ R 2 ), p ∈ Z 2 in Definition 5.1, according to the point p ∈ Z 2 as stated in (5.1), where the point p of (1) is a doubly even point, the point p of (2) is an even point, and each of the points p of (3) and (4) is an odd point.
A p by taking the following way.
A p = R 2 by taking the following way.
(2) In case Y is a disconnected subset of (Z 2 , γ), A(Y) may not be a disconnected subset of (R 2 , U). Namely, the connectedness of A(Y) in (R 2 , U) depends on the situation.
Proof: (1) Given a connected subset X of (Z 2 , γ), we obtain A(X) = p∈X A p that is a connected subset of (R 2 , U) (see Definition 5.1).
(2) As an example, consider the set {p, q}, where p, q ∈ (Z 2 ) e , p q, and q ∈ N 8 (p). While the set {p, q} is a disconnected subset of (Z 2 , γ), A({p, q}) = A p ∪ A q is a connected subset of (R 2 , U) (see Definition 5.1). For instance, in Figure 4(1), consider the two points p := (0, 0) and q := (1, −1) in S C 4 γ . Then the set {p, q} supports the assertion.  Figure 4(1)(a),(b)). Then it is clear that A(S C 4 γ ) does not separate (R 2 , U) into two nonempty components. (2) In the case of l ∈ {8, 10}, we obtain A(S C l γ ) as a subspace of (R 2 , U) whose complement of A(S C l γ ) in R 2 consists of only two non-empty components. See the process of obtaining A(S C 8 γ ) in Figure 4(2) from (a) to (b) and (c) to (d).
(3) For any l of S C l γ , l {4, 8, 10}, A(S C l γ ) separates (R 2 , U) to obtain that R 2 \ A(S C l γ ) has more than or equal to two non-empty components. For instance, as shown in each of the objects S C 12 γ in Figure  4(4)(a),(b), we see that R 2 \ A(S C 12 γ ) has only two non-empty components. Meanwhile, as for the S C 12 γ as in Figure 4(c) and S C 18 γ in Figure 5(a), each of R 2 \ A(S C 12 γ ) and R 2 \ A(S C 18 γ ) has more than two components. To be specific, we need to strongly recognize that the set R 2 \ A(S C 12 γ ) of Figure 4(4)(b) has only two components, i.e., we have C(p 1 ) = C(p 2 ), where C(p i ) means the component containing the given point p i in (R 2 , U), i ∈ {1, 2}. Similarly, R 2 \ A(S C 18 γ ) of Figure 5(d) has only two components. However, as shown in Figure 4(4) from (c) to (d), we see that R 2 \ A(S C 12 γ ) has three disjoint non-empty components such as C(q 1 ), C(q 2 ), and R 2 \ (C(q 1 ) ∪ C(q 2 ) ∪ A(S C 12 γ )).
Theorem 5.7. Assume an S C l γ := (d i ) i∈[0,l−1] Z , l 4, satisfying the following property: (1) Then A(S C l γ ) separates (R 2 , U) into exactly two components in (R 2 , U) that are both semi-open and semi-closed.
(2) One of the components of R 2 \ A(S C l γ ) is bounded and the other is unbounded in R 2 .
Before proving the assertion, we need to recognize that the hypothesis requires that S C l γ always satisfies the property of (5.2). In particular, we need to focus on the part "S C l γ ∩ (Z 2 ) e " of (5.2).
Proof: Owing to the hypothesis, assume an S C l γ := (d i ) i∈[0,l−1] Z , l 4, that does not contain the subset having the following property: where Con(d t 1 ) is the connected maximal subset of N 8 (d t 1 ) ∩ S C l γ containing the point d t 1 . Then, owing to the notion of (5.1) and the features of S C l γ , A(S C l γ ) separates (R 2 , U) into exactly two both semi-open and semi-closed components in (R 2 , U).
For instance, consider the case of S C 12 γ given in Figure 4(4)(a). Then we obtain A(S C 12 γ ) to separate (R 2 , U) into exactly two both semi-open and semi-closed components in (R 2 , U). More precisely, as in Figure 4(4)(a),(b), owing to the property of (5.1), the set A p 1 ∪ A p 2 is a connected subset of (R 2 , U).
Meanwhile, without the hypothesis, we can consider the following case.
Assume an S C l γ := (d i ) i∈[0,l−1] Z , l 4, in which there is a subset X 1 (⊂ S C l γ ∩ (Z 2 ) e ) of (5.3) such that d t 2 ∈ N 8 (d t 1 ) and Con(d t 1 ) ∩ {d t 2 } = ∅. To be specific, see the two points r 1 and r 2 in Figure 5(a). Then, A(S C l γ ) does not separate (R 2 , U) into exactly two both semi-open and semi-closed components in (R 2 , U). As another example, see the objects in Figure 4(4)(c),(d). To be specific, in Figure 4(4)(c),(d), the set A q 1 is not connected with A q 2 (see the property of (5.1)).
(2) With the hypothesis, it is clear that one of the components of R 2 \ A(S C l γ ) is bounded and the other is unbounded in R 2 .
Theorem 5.8. Assume the subspaces (X, γ X ) and (Y, γ Y ) that are MW-homeomorphic to S C l γ . Then the number of the components of R 2 \ A(X) need not be equal to that of R 2 \ A(Y).
Proof: It suffices to prove it by suggesting a counterexample. Given the two types of S C 18 γ as in Figure 5(a),(c), we obtain the corresponding two types of A(S C 18 γ ) as in Figure 5(b),(d). For our purpose, let A 1 (resp. A 2 ) be the set R 2 \ A(S C 18 γ ), where S C 18 γ is the object in Figure 5(a) (resp. Figure 5(c)). Then it is clear that A 1 has four components and A 2 has the only two components as in Figure 5(b),(d).
Remark 5.9. The number the components of R 2 \ A(S C l γ ) need not be equal to that of Z 2 \ S C l γ . Proof: As a counterexample, see the objects in Figure 5 the set R 2 \ A(S C 18 γ ) has exactly two components as in Figure 5(d) because C(q 1 ) = C(q 2 ) = C(q 3 ) as the subspaces of (R 2 , U), i.e., C(q 1 ) is connected with C(q 2 ) and C(q 2 ) is also connected with C(q 3 ). To be specific, see the cases of S C 18 γ in Figure 5(c) and A(S C 18 γ ) in Figure 5(d) stated in Example 5.2, which completes the proof.
Then the number of the components of R 2 \ A(S C l γ ) in (R 2 , U) is equal to that of Z 2 \ S C l γ in (Z 2 , γ). Proof: First of all, without the hypothesis, we need to show that the assertion does not hold. Suppose an S C l γ := (c i ) i∈[0,l−1] Z , l {4, 8, 10}, that does not satisfy the property of (5.4). For instance, as shown in Figure 5(c),(d), the given S C 18 γ in Figure 5(c) does not satisfy the property of (5.4), i.e., see the two points s 1 and s 2 in Figure 5(c). Based on the S C 18 γ , we find that R 2 \ A(S C 18 γ ) has only two components in (R 2 , U) (see Example 5.2). Meanwhile, Z 2 \ S C 18 γ has four components in (Z 2 , γ). Next, owing to the property of (5.1), the topological feature of R 2 \ A(S C l γ ) in (R 2 , U), and that of Z 2 \ S C l γ in (Z 2 , γ), the proof is completed. Example 5.3. Let us consider S C 18 γ in Figure 5(a). Then it is clear that it satisfies the property of (5.4) so that we obtain R 2 \ A(S C 18 γ ) consisting of four components in (R 2 , U) (see Figure 5(b)) and Z 2 \ S C 18 γ is composed of four components in (Z 2 , γ) (see Figure 5(a)).

Semi-Jordan curve theorem on the MW-topological plane
In 1970, Rosenfeld [9][10][11] initially considered the digital topological version of the typical Jordan curve theorem (see also [22]). Consider S C 2,l k in a binary digital picture D := (Z 2 , k,k, S C 2,l k ). Then thek-components are called white components of D and S C 2,l k is said to be a black component (or equivalently, k-component) of the digital picture [22], where we say that a k-component of a nonempty digital image (X, k) is a maximal k-connected subset of (X, k) [22]. To be precise, given an S C 2,4 8 on Z 2 , to evade from the so-called "digital connectivity paradox" [10,11], the papers [7][8][9] Meanwhile, unlike this approach followed from Rosenfeld's work, in the category of MW-topological spaces, given an S C l γ in (Z 2 , γ), we now raise the following queries. (Q1) How can we propose an MW-topological version of the typical Jordan curve theorem? (Q2) What differences are there between an MW-topological version of the well-known Jordan curve theorem and the typical Jordan curve theorem on (R 2 , U) ? (Q3) What differences are there between the Jordan curve theorem in an MW-topological setting and the digital Jordan curve theorem established by Rosenfeld? The paper [12] also studied several types of digital Jordan curve theorems with nine pretopologies on Z 2 . Besides, the paper [3] also proposed a computational topological version of the curve and surface theorem. In addition, there are some studies on the digital versions of the Jordan curve theorem in digital spaces including the papers [6,12,22]. However, to study this topic more intensively from the viewpoint of the MW-topology, we strongly need to have an approach using semi-topological structures. To study some properties of the semi-closedness or semi-openness of Z 2 \S C l γ , we first recall that S C 4 γ does not separate (Z 2 , γ) into exactly two non-empty components (see Figure 1(1)(a),(b)). Furthermore, we have the following (see Figure 4(1)(a), (2)(a), (3)(a)).
γ has the only one non-empty component that is both semi-open and semi-closed in (Z 2 , γ).
γ separates (Z 2 , γ) into exactly two semi-closed components. One of these components need not be semi-open in (Z 2 , γ).
(2) S C 10 γ separates (Z 2 , γ) into exactly two components that are both semi-open and semi-closed. Proof: Based on the S C 8 γ in Figure 1(2)(a),(b), and S C 10 γ in Figure 1(3)(a),(b), by Theorems 3.3 and 3.4, the proof is clearly completed. In particular, note that the components of Z 2 \ S C 8 γ are obviously semi-closed in (Z 2 , γ). Indeed, in the case of S C 8 γ in Figure 1(2)(a), one of them is not semi-open. To be specific, consider S C 8 γ in Figure 1 There are no distinct elements c t 1 , c t 2 in S C l γ such that c t 2 ∈ N 8 (c t 1 ) and Con(c t 1 ) ∩ {c t 2 } = ∅, where Con(c t 1 ) is the connected maximal subset of N 8 (c t 1 ) ∩ S C l γ containing the point c t 1 .
Then S C l γ separates (Z 2 , γ) into exactly two semi-closed components, e.g., A and B. Namely, a partition {A, B, S C l γ } of (Z 2 , γ) exists. Proof: First of all, we need to strongly point out an importance of the given hypothesis. Without the hypothesis, as shown in Figure 4(4)(a),(c), since the given S C 12 γ in Figure 4(4)(a),(c) do not satisfy the hypothesis of (6.3), they do not separate (Z 2 , γ) into exactly two components. Besides, some components of Z 2 \ S C 12 γ cannot be semi-open. To be specific, in the S C 12 γ of Figure 4(4)(a), each of C(p i ) = {p i }, i ∈ {1, 2}, is not semi-open in (Z 2 , γ).
As another case, since the given S C 12 γ in Figure 4(4)(c) does not satisfy the property of (6.3) either, it does not separate (Z 2 , γ) into exactly two components. Besides, note that none of the objects in Figure 5(a),(c), Figure 6, and Figure 7(1),(2) satisfies the property of (6.3) either.
In addition, the condition of (6.3) does not support the semi-openness of the component of Z 2 \ S C l γ .
For instance, consider S C 8 γ in Figure 1(2)(a), one component of Z 2 \ S C 8 γ is not semi-open in (Z 2 , γ) as mentioned in the proof of Lemma 6.2.
Meanwhile, with the hypothesis of (6.3), owing to the features of S C l γ , it is clear that S C l γ separates (Z 2 , γ) into exactly two components (see Figure 1(4)(c),(d)) which are semi-closed in (Z 2 , γ).
Owing to Remark 6.1, based on Theorem 5.7, we can define the notions of Definition 6.4 below because given an S C l γ satisfying the hypothesis of (5.2), R 2 \ A(S C l γ ) has exactly two components in (R 2 , U) and further, one of them is bounded and the other is unbounded (see the cases of S C 38 γ in Figure 7(1) and S C 28 γ in Figure 7(3)).
Definition 6.4. Assume an S C l γ satisfying the property of (5.2), l 4. Then we define the following two notions.
Comparing the condition of Proposition 6.3 and that of Definition 6.4, we can note that the former is stronger than the latter. Hereinafter, the two notions I(S C l γ ) and O(S C l γ ) in Definition 6.4 are called "inside" and "outside" of S C l γ in (Z 2 , γ), respectively. In particular, note that these notions are not related to the notions of interior and exterior of a set of (Z 2 , γ). Remark 6.5. (1) In Definition 6.4, the hypothesis of (5.2) is strongly required to establish both I(S C l γ ) and O(S C l γ ) because it supports the assertion of Theorem 5.7 so that the set R 2 \ A(S C l γ ) has exactly two components of which one of them is bounded and the other is unbounded.
(2) Without the hypothesis of (5.2), we have some difficulties in establishing the notions of I(S C l γ ) and O(S C l γ ). For instance, consider the case of S C 42 γ in Figure 6. In particular, see the points c 2 and c 38 , and c 4 and c 36 . Then, owing to these points, it is clear that this S C 42 γ does not satisfy the property of (5.2). Hence we have some difficulties in establishing I(S C 42 γ ) because R 2 \ A(S C 42 γ ) does not have exactly two components. As another case, consider the case of S C 28 γ in Figure 7(2). In particular, consider the two points c 19 and c 25 . Then they clearly does not satisfy the property of (5.2) so that R 2 \ A(S C 28 γ ) has three components, e.g., two bounded components and one unbounded component. More precisely, since the set A q 2 is a bounded component of R 2 \ A(S C 28 γ ), A q 2 is not related to the set Ub(R 2 \ A(S C 28 γ )) ∩ Z 2 which comes across some difficulties in establishing O(S C 28 γ ). (3) As a good example for Definition 6.4, consider the S C 38 γ in Figure 7(1). First, see the two points d 29 and d 35 of the S C 38 γ in Figure 7(1). Then they can be admissible to establish O(S C 38 γ ). Besides, see the two points d 11 and d 17 of the S C 38 γ in Figure 7(1). Then they can be admissible to establish the notion of I(S C 38 γ ) because A q 1 is connected with A q 2 . Similarly, the S C 28 γ in Figure 7(3) also a good example for Definition 6.4.
(1) As for the S C 42 γ in Figure 6, it does not satisfy the property of (5.2) (see the points c 2 and c 38 , and c 4 and c 36 ). Hence, as mentioned in Theorem 5.7, A(S C 42 γ ) does not separate (R 2 , U) into two components. Indeed, R 2 \ A(S C 42 γ ) consists of four components as follows: C(p i ) = A p i , i ∈ [1, 2] Z , C(p 3 ) = C(p 4 ) and C(q 1 ) = C(q 2 ). Furthermore, the set Z 2 \ S C 42 γ has six components such as C(p i ), i ∈ [1, 4] Z and C(q j ), j ∈ [1, 2] Z . (2) Given the S C 38 γ in Figure 7(1), it satisfies the property of (5.2). Hence the set R 2 \ A(S C 38 γ ) consists of the exactly two components. Hence we obtain Meanwhile, the set Z 2 \ S C 38 γ consists of four components in (Z 2 , γ), e.g., C(q i ), i ∈ [1, 4] Z . (3) Given the S C 28 γ in Figure 7(2), it does not satisfy the property of (5.2). Indeed, the set Z 2 \ S C 28 γ consists of three components as follows: C(q i ), i ∈ [1, 3] Z and C(q 1 ) consists of eleven elements as in Figure 7(2). In particular, C(q 2 ) = {q 2 } and (C(q 3 )) = ℵ 0 that is the cardinal number of the set of natural numbers.
(4) Given the S C 28 γ in Figure 7(3), it is clear that this object satisfies the hypothesis of Definition 6.4. Hence we obtain I(S C 28 γ ) = C(q 1 ) and O(S C 28 γ ) = C(q 2 ) ∪ C(q 3 ).  Figure 6. In (Z 2 , γ), based on the non-satisfaction of the property of (5.2) of S C 42 γ , there are some difficulties in establishing I(S C 42 γ ).
Owing to Definition 6.4, we have the following: Theorem 6.6. Assume an S C l γ , l 4, satisfying the property of (5.2). Then, a partition {I(S C l γ ), O(S C l γ ), S C l γ } of Z 2 exists.  Figure 7(2), Z 2 \ S C 28 γ indeed has three components. However, since it does not satisfy the property of (5.2), both I(S C 28 γ ) and O(S C 28 γ ) are not considered.
Proof: Owing to Definition 6.4, the proof is completed.  Figure 7(1). It is clear that the given S C 38 γ satisfies the hypothesis of Definition 6.4. While each of I(S C 38 γ ) and O(S C 38 γ ) exists, they are not connected in (Z 2 , γ). To be specific, it turns out that such that C(q 1 ) ∩ C(q 2 ) = ∅, C(q 3 ) ∩ C(q 4 ) = ∅, and each of C(q i ), i ∈ [1, 4] Z is not an empty set.
(2) From the above (3) of Example 6.1, it is clear that the number of the components of I(S C l γ ) depends on the number l.
(3) As an example, consider the S C 8 γ in Figure 4(2)(a). Then I(S C 8 γ ) is not an open set. Besides, consider the S C 38 γ in Figure 7(1). Then O(S C 38 γ ) is not an open set. Let us now investigate the semi-openness or semi-closedness of I(S C l γ ) and O(S C l γ ).
Example 6.2. Given any S C 8 γ , there is a partition of (Z 2 , γ), i.e., {I(S C 8 γ ), S C 8 γ , O(S C 8 γ )} such that each of I(S C 8 γ ) and O(S C 8 γ ) is semi-closed and both of them are connected. Namely, S C 8 γ separates (Z 2 , γ) with exactly two components.  Proof: The semi-topological features of I(S C 8 γ ) is determined according to the two types of S C 8 γ in Figure 1(2)(a),(b). To be specific, based on the S C 8 γ in Figure 1(2)(a), we have I(S C 8 γ ) that is only semi-closed instead of semi-open in (Z 2 , γ). Meanwhile, for the case of S C 8 γ in Figure 1(2)(b), I(S C 8 γ ) is proved to be both semi-open and semi-closed in (Z 2 , γ). Next, let us consider the case S C l γ , 12 ≤ l ∈ N e . Then the semi-openness or semi-closedness of I(S C l γ ) depends on the situation (see Figure 5(c) and 7(1), (3)). For instance, the S C 18 γ in Figure 5(c) has I(S C 18 γ ) that is not semi-open but semi-closed.
Corollary 6.10. Let X and Y be simple closed MW-curves with l elements in (Z 2 , γ) and each of them satisfies the property of (6.3). Then the number of the components of X c is equal to that of Y c .
Proof: By Proposition 6.3, the proof is completed. Let E be the S C 28 γ in Figure 7(3) and F be the S C 28 γ satisfying the property of (6.3). While O(E) is not connected and O(F) is connected. Thus we obtain the following. Remark 6.11. Without the condition relating to the property (6.3), each of I(S C l γ ) and O(S C l γ ) may not be connected. Theorem 6.12. Assume that the subspaces (X, γ X ) and (Y, γ Y ) are MW-homeomorphic to S C l γ . (1) (I(X), γ I(X) ) need not MW-homeomorphic to (I(Y), γ I(X) ).
Proof: To disprove these assertions (1) and (2), we will use some examples.
(1) Consider the two S C 12 γ in Figures 4(4)(a) and 5(a). For our purpose, let A be the S C 12 γ in Figure  4(4)(a) and B be the S C 12 γ in Figure 4(5)(a). While I(A) is not connected and I(B) is connected, which completes the proof.
(2) For our purpose, let C be the S C 28 γ in Figure 7(3) and D be the S C 28 γ satisfying the property of (6.3). While O(C) is not connected and O(D) is connected, which completes the proof. Proof: To disprove these assertions (1) and (2), we will use some examples. (1) Let us consider the S C 28 γ in Figure 7(3). Then it is clear that O(S C 28 γ ) is not semi-open. (2) In view of the S C 28 γ in Figure 7(3), the proof is completed. To be specific, let A be the S C 28 γ in Figure 7(3) and B be the S C 28 γ satisfying the property of (6.3). Then O(A) has two components and O(B) has the only one component.
Unlike Theorem 6.6, let us find a condition to separate (Z 2 , γ) into exactly two components, as follows: Theorem 6.14. Let S C l γ satisfy the property of (6.3), l 4. Under (Z 2 , γ), we obtain the following: Proof: (1) It suffices to prove the connectedness of both I(S C l γ ) and O(S C l γ ). Owing to the hypothesis of the property of (6.3), each of I(S C l γ ) and O(S C l γ ) is proved to be connected because there are not two points        c t 1 , c t 2 ∈ S C l γ := (c i ) i∈[0,l−1] Z , l 4 such that c t 2 ∈ N 8 (c t 1 ) and Con(c t 1 ) ∩ {c t 2 } = ∅.
Indeed, this property of (6.4) supports the connectedness of both I(S C l γ ) and O(S C l γ ). (2) With the hypothesis, using Theorems 3.3 and 3.4, we prove that both I(S C l γ ) and O(S C l γ ) are semiclosed. To be specific, for any point x ∈ Z 2 \ I(S C l γ ), owing to the hypothesis, we always obtain S N γ (x) ∩ (Z 2 \ I(S C l γ )) op ∅, which implies the semi-closedness of I(S C l γ ). Similarly, for any point x ∈ Z 2 \ O(S C l γ ), owing to the hypothesis, we also obtain S N γ (x) ∩ (Z 2 \ O(S C l γ )) op ∅, which implies the semi-closedness of O(S C l γ ). Besides, by Theorem 3.3, owing to the hypothesis, O(S C l γ ) is proved to be semi-open. However, I(S C l γ ) need not be semi-open. For instance, for the S C 8 γ in Figure 4(2)(a). Then I(S C 8 γ ) is not a semi-open in (Z 2 , γ).
7. Advantages and utilities of MW-topological structure and the semi-Jordan curve theorem on (Z 2 , γ) When studying digital objects X in Z 2 , the properties of (2.4) and (2.5) enable us to get the following utilities of the MW-topological structure of X.
Since the modern electronic devices are usually operated on the finite digital planes with more than ten million pixels to support the high-level display resolution, the mapping of (7.1) can be very admissible. At the moment, note that the following map g cannot be a homeomorphism, where (2) Since the MW-topological structure is one of the fundamental frames, motivated by this structure, some more generalized topological structures on Z n can be established.
(3) Based on the MW-topological structure of Z 2 , we can obtain the 4-digital adjacency induced by the given topological structure [26]. In detail, for distinct elements x, y ∈ (Z 2 , γ), they are MW-adjacent if x ∈ S N γ (y) or y ∈ S N γ (x) [26]. Namely, the MW-adjacency is equivalent to the 4-adjacency of Z 2 as in (2.2).
Remark 7.2. (Advantages of the semi-Jordan curve theorem) (1) Unlike the typical Jordan curve theorem in a digital topological setting estabsihed by Rosenfeld [22], no paradox exists in the semi-Jordan curve theorem in the MW-topological structure.
(2) Based on the semi-Jordan curve theorem in the MW-topological structure, we can consider a digital topological version of the typical Jordan curve theorem in terms of a simple closed 4-curve in the digital plane (Z 2 , 4, 8).
(3) When digitizing a set X in the 2-dimensional real space with respect to the MW-topological structure, we can use a local rule in [20] to obtain a digitized set D γ (X) ⊂ Z 2 from X which is used in the fields of mathematical morphology, rough set theory, digital geometry, and so on [20,21].

Concluding remark and further work
After developing the semi-Jordan curve theorem in the MW-topological setting, we have studied various properties of it. In particular, we have found a condition for S C l γ to separate (Z 2 , γ) with exactly two components, Furthermore, we studied many semi-topological properties of both I(S C l γ ) and O(S C l γ ). As a further work, we can compare among several kinds of digital versions of the typical Jordan curve theorem and the combinatorial version of the Jordan curve theorem in [3]. Besides, based on the digital-topological group structure in [32], we can further examine a topological group structure of S C l γ . In addition, based on the established structure in [33], we can study covering spaces in the category of MW-topological spaces.