Semi-Jordan curve theorem on the Marcus-Wyse topological plane
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The paper initially develops the semi-Jordan curve theorem on the digital plane with the Marcus-Wyse topology, i.e., $ MW $-topological plane or $ ({\mathbb Z}^2, \gamma) $ for brevity. We first prove that while every simple closed $ MW $-curve is semi-open in $ ({\mathbb Z}^2, \gamma) $, it may not be semi-closed. Given a simple closed $ MW $-curve with $ l $ elements, denoted by $ SC_{\gamma}^l $, after establishing a continuous analog of $ SC_{\gamma}^l $ denoted by $ \mathcal{A}(SC_{\gamma}^l) $, we initially show that $ \mathcal{A}(SC_{\gamma}^l) $ is both semi-open and semi-closed in $ ({\mathbb R}^2, \mathcal{U}) $, where $ ({\mathbb R}^2, \mathcal{U}) $ is the $ 2 $-dimensional real plane $ {\mathbb R}^2 $ with the usual topology $ \mathcal{U} $. Furthermore, we find a condition for $ \mathcal{A}(SC_{\gamma}^l) $ to separate $ ({\mathbb R}^2, \mathcal{U}) $ into exactly two non-empty components, compared to a typical Jordan curve theorem on $ ({\mathbb R}^2, \mathcal{U}) $. Since not every $ SC_{\gamma}^l $ always separates ($ {\mathbb Z}^2, \gamma) $ into two nonempty components, we find a condition for $ SC_{\gamma}^l, l\neq 4, $ to separate $ ({\mathbb Z}^2, \gamma) $ into exactly two components. The semi-Jordan curve theorem on the $ MW $-topological plane plays an important role in applied topology such as digital topology, mathematical morphology as well as computer science.
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Keywords:
- semi-Jordan curve theorem,
- semi-open,
- semi-closed,
- Alexandroff space,
- Marcus-Wyse topology,
- Marcus-Wyse ($ MW $-, for brevity) topological plane,
- semi-homeomorphism,
- continuous analog of a digital object,
- digital-topological group,
- digital topology
Citation: Sang-Eon Han. Semi-Jordan curve theorem on the Marcus-Wyse topological plane[J]. Electronic Research Archive, 2022, 30(12): 4341-4365. doi: 10.3934/era.2022220
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Abstract
The paper initially develops the semi-Jordan curve theorem on the digital plane with the Marcus-Wyse topology, i.e., $ MW $-topological plane or $ ({\mathbb Z}^2, \gamma) $ for brevity. We first prove that while every simple closed $ MW $-curve is semi-open in $ ({\mathbb Z}^2, \gamma) $, it may not be semi-closed. Given a simple closed $ MW $-curve with $ l $ elements, denoted by $ SC_{\gamma}^l $, after establishing a continuous analog of $ SC_{\gamma}^l $ denoted by $ \mathcal{A}(SC_{\gamma}^l) $, we initially show that $ \mathcal{A}(SC_{\gamma}^l) $ is both semi-open and semi-closed in $ ({\mathbb R}^2, \mathcal{U}) $, where $ ({\mathbb R}^2, \mathcal{U}) $ is the $ 2 $-dimensional real plane $ {\mathbb R}^2 $ with the usual topology $ \mathcal{U} $. Furthermore, we find a condition for $ \mathcal{A}(SC_{\gamma}^l) $ to separate $ ({\mathbb R}^2, \mathcal{U}) $ into exactly two non-empty components, compared to a typical Jordan curve theorem on $ ({\mathbb R}^2, \mathcal{U}) $. Since not every $ SC_{\gamma}^l $ always separates ($ {\mathbb Z}^2, \gamma) $ into two nonempty components, we find a condition for $ SC_{\gamma}^l, l\neq 4, $ to separate $ ({\mathbb Z}^2, \gamma) $ 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.
References
[1] J. R. Munkres, Topology A First Course, Prentice-Hall, Inc., 1975. [2] D. Fajardo-Rojas, N. Jonard-Pérez, A Jordan curve theorem for $2$-dimensional tillings, Topol. Appl., 300 (2021), 107773. https://doi.org/10.1016/j.topol.2021.107773 doi: 10.1016/j.topol.2021.107773 
[3] S. E. Han, Jordan surface theorem for simple closed $SST$-surfaces, Topol. Appl., 272 (2020), 106953. https://doi.org/10.1016/j.topol.2019.106953 doi: 10.1016/j.topol.2019.106953
[4] G. T. Herman, Oriented surfaces in digital spaces, CVGIP: Graphical Models Image Proc., 55 (1993), 381–396. https://doi.org/10.1006/cgip.1993.1029 doi: 10.1006/cgip.1993.1029
[5] E. Khalimsky, R. Kopperman, P. R. Meyer, Computer graphics and connected topologies on finite ordered sets, Topol. Appl., 36 (1990), 1–17. https://doi.org/10.1016/0166-8641(90)90031-V doi: 10.1016/0166-8641(90)90031-V
[6] C. O. Kiselman, Digital Jordan curve theorem, in International Conference on Discrete Geometry for Computer Imagery, (2000), 46–56. https://doi.org/10.1007/3-540-44438-6_5 [7] A. Rosenfeld, Connectivity in digital pictures, J. Assoc. Comput. Math., 17 (1970), 146–160. https://doi.org/10.1145/321556.321570 doi: 10.1145/321556.321570
[8] A. Rosenfeld, Arcs and curves in digital pictures, J. Assoc. Comput. Math., 20 (1973), 81–87. https://doi.org/10.1145/321738.321745 doi: 10.1145/321738.321745
[9] A. Rosenfeld, Adjacency in digital pictures, Inf. Control, 26 (1974), 24–33. https://doi.org/10.1016/S0019-9958(74)90696-2 doi: 10.1016/S0019-9958(74)90696-2
[10] A. Rosenfeld, A converse to the Jordan curve theorem for digital curves, Inf. Control, 29 (1975), 292–293. https://doi.org/10.1016/S0019-9958(75)90459-3 doi: 10.1016/S0019-9958(75)90459-3
[11] A. Rosenfeld, Digital topology, Am. Math. Monthly, 86 (1979), 76–87. https://doi.org/10.1080/00029890.1979.11994873 [12] J. ${\check S}$lapal, A quotient-universal digital topology, Theor. Comput. Sci., 405 (2008), 164–175. https://doi.org/10.1016/j.tcs.2008.06.035 doi: 10.1016/j.tcs.2008.06.035
[13] N. Levine, Semi-open sets and semi-continuity in topological spaces, Am. Math. Monthly, 70 (1963), 36–41. https://doi.org/10.2307/2312781 doi: 10.2307/2312781
[14] N. Levine, Generalized closed sets in topology, Rend. Circ. Mat. Palermo, 19 (1970), 89–96. https://doi.org/10.1007/BF02843888 doi: 10.1007/BF02843888
[15] S. G. Crosseley, A note on semitopological properties, Proc. Am. Math. Soc., 72 (1978), 409–412. https://doi.org/10.1090/S0002-9939-1978-0507348-9 doi: 10.1090/S0002-9939-1978-0507348-9
[16] T. Hamlett, A correction to the paper "Semi-open sets and semi-continuity in topological spaces" by Norman Levine, Proc. Am. Math. Soc., 49 (1975), 458–460. https://doi.org/10.2307/2040665 doi: 10.2307/2040665
[17] H. Maki, K. Balachandran, R. Devi, Remarks on semi-generalized closed sets and generalized semi-closed sets, Kyungpook Math. J., 36 (1996), 155–163. [18] A. S. Mashhour, M. E. Abd El-Monsef, S.N. El-Deeb, On precontinuous an weak precontinuous mappings, Proc. Math. Phys. Soc. Egypt, 53 (1982), 47–53. [19] T. Noiri, A note on semi-homeomorphism, Bull. Calcutta Math. Soc., 76 (1984), 1–3. [20] S. E. Han, Roughness measures of locally finite covering rough sets, Int. J. Approx. Reason., 105 (2019), 368–385. https://doi.org/10.1016/j.ijar.2018.12.003 doi: 10.1016/j.ijar.2018.12.003
[21] S. E. Han, W. Yao, Homotopy based on Marcus-Wyse topology and its applications, Topol. Appl., 201 (2016), 358–371. https://doi.org/10.1016/j.topol.2015.12.047 doi: 10.1016/j.topol.2015.12.047
[22] T. Y. Kong, A. Rosenfeld, Topological Algorithms for the Digital Image Processing, Elsevier Science, Amsterdam, 1996. [23] S. E. Han, Non-product property of the digital fundamental group, Inf. Sci., 171 (2005), 73–91. https://doi.org/10.1016/j.ins.2004.03.018 doi: 10.1016/j.ins.2004.03.018
[24] P. Alexandorff, Uber die Metrisation der im Kleinen kompakten topologischen Räume, Math. Ann., 92 (1924), 294–301. https://doi.org/10.1007/BF01448011 doi: 10.1007/BF01448011
[25] P. Alexandorff, Diskrete Räume, Mat. Sb., 2 (1937), 501–518. [26] S. E. Han, Generalizations of continuity of maps and homeomorphisms for studying $2$D digital topological spaces and their applications, Topol. Appl., 196 (2015), 468–482. https://doi.org/10.1016/j.topol.2015.05.024 doi: 10.1016/j.topol.2015.05.024
[27] D. Marcus, F. Wyse, Solution to problem 5712, Am. Math. Monthly, 77 (1970), 1119. https://doi.org/10.2307/2316121 [28] S. E. Han, Continuities and homeomorphisms in computer topology and their applications, J. Korean Math. Soc., 45 (2008), 923–952. https://doi.org/10.4134/JKMS.2008.45.4.923 doi: 10.4134/JKMS.2008.45.4.923
[29] S. G. Crosseley, S. K. Hildebrand, Semi-closure, Texas. J. Sci., 22 (1971), 99–112. [30] S. E. Han, Low-level separation axioms from the viewpoint of computational topology, Filomat, 33 (2019), 1889–1901. https://doi.org/10.2298/FIL1907889H doi: 10.2298/FIL1907889H
[31] S. E. Han, Semi-topological properties of the Marcus-Wyse topological spaces, AIMS Math., 7 (2022), 12742–12759. https://doi.org/10.3934/math.2022705 doi: 10.3934/math.2022705
[32] S. E. Han, Digitally topoloigical groups, Electron. Res. Arch., 30 (2022), 2356–2384. https://doi.org/10.3934/era.2022120 doi: 10.3934/era.2022120
[33] S. E. Han, The most refined axiom for a digital covering space, Mathematics, 8 (2020), 1868. https://doi.org/10.3390/math8111868 doi: 10.3390/math8111868
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- Figure 1. Examples of several types of $ SC_{\gamma}^l $, where $ l \in \{4, 8, 10, 12\} $. Namely, (1) $ SC_{\gamma}^4 $ is both semi-open and semi-closed in $ ({\mathbb Z}^2, \gamma) $. (2) As for $ SC_{\gamma}^8 $, while the object of (a) is not semi-closed but semi-open, the object of (b) is both semi-closed and semi-open in $ ({\mathbb Z}^2, \gamma) $. (3) As for $ SC_{\gamma}^{10} $, the objects of (a) and (b) are both semi-open and semi-closed in $ ({\mathbb Z}^2, \gamma) $. (4) As for $ SC_{\gamma}^{12} $, while the object of (a) is not semi-closed but semi-open, each of (b)–(d) is both semi-open and semi-closed in $ ({\mathbb Z}^2, \gamma) $
- Figure 2. In $ ({\mathbb Z}^2, \gamma) $, consider the $ SC_{\gamma}^{12} $ in Figure 2(a). As shown in Figure 2(b), owing to the two points $ p_1, p_2 $ in $ {\mathbb Z}^2 \setminus SC_{\gamma}^{12} $ in Figure 2(a), we conclude that $ SC_{\gamma}^{12} $ is not semi-closed in $ ({\mathbb Z}^2, \gamma) $ because $ Int(Cl(SC_{\gamma}^{12}))\nsubseteq SC_{\gamma}^{12} $ (see the points $ p_1 $ and $ p_2 $ in (b)). However, we obtain $ SC_{\gamma}^{12} \subset Cl(Int(SC_{\gamma}^{12})) $ that implies the semi-openness of $ SC_{\gamma}^{12} $
- Figure 3. Configurations of $ A_p (\subset {\mathbb R}^2), p \in {\mathbb Z}^2 $ in Definition 5.1, according to the point $ p \in {\mathbb 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
- Figure 4. Given several types of $ SC_{\gamma}^l $, configuration of $ \mathcal{A}(SC_{\gamma}^l) $ according to the given $ SC_{\gamma}^l, l \in \{4, 8, 10, 12\} $ mentioned in Proposition 5.6. More precisely, (1) $ SC_{\gamma}^4 \to \mathcal{A}(SC_{\gamma}^4) $ (2) $ SC_{\gamma}^8 \to \mathcal{A}(SC_{\gamma}^8) $ (3) $ SC_{\gamma}^{10} \to \mathcal{A}(SC_{\gamma}^{10}) $ (4) Two types of processes for obtaining $ \mathcal{A}(SC_{\gamma}^{12}) $ from the non-semi-closed $ SC_{\gamma}^{12} $ of (a) and the semi-closed $ SC_{\gamma}^{12} $ of (c) (5) $ SC_{\gamma}^{12} \to \mathcal{A}(SC_{\gamma}^{12}) $
- Figure 5. Assume the two types of $ SC_{\gamma}^{18} $ in (a) and (c) in Example 5.2. The set in (b) is the set $ \mathcal{A}(SC_{\gamma}^{18}) $ obtained from the object of (a) and the set $ \mathcal{A}(SC_{\gamma}^{18}) $ in (d) is derived from the object of (c)
- Figure 6. In $ ({\mathbb Z}^2, \gamma) $, based on the non-satisfaction of the property of (5.2) of $ SC_{\gamma}^{42} $, there are some difficulties in establishing $ I(SC_{\gamma}^{42}) $
- Figure 7. In $ ({\mathbb Z}^2, \gamma) $, (1) based on the $ SC_{\gamma}^{38}: = (d_i)_{i \in [0, 37]_{\mathbb Z}} $ satisfying the property of (5.2), $ {\mathbb Z}^2 \setminus SC_{\gamma}^{38} $ has four non-empty components which implies that $ I(SC_{\gamma}^{38}) = C(q_1) \cup C(q_2) $ and $ O(SC_{\gamma}^{38}) = C(q_3) \cup C(q_4) $ such that $ C(q_1) \cap C(q_2) = \emptyset $ and $ C(q_3) \cap C(q_4) = \emptyset $, where $ C(q_4) = {\mathbb Z}^2 \setminus (I(SC_{\gamma}^{38})\cup C(q_3) \cup SC_{\gamma}^{38}) $ and each of $ C(q_i), i \in [1,4]_{\mathbb Z} $, is not an empty set. (2) Based on the $ SC_{\gamma}^{28}: = (c_i)_{i \in [0, 27]_{\mathbb Z}} $ in Figure 7(2), $ {\mathbb Z}^2 \setminus SC_{\gamma}^{28} $ indeed has three components. However, since it does not satisfy the property of (5.2), both $ I(SC_{\gamma}^{28}) $ and $ O(SC_{\gamma}^{28}) $ are not considered