The present paper aims to investigate some semi-separation axioms relating to the Alexandroff one point compactification (Alexandroff compactification, for short) of the digital plane with the Marcus-Wyse ($ MW $-, for brevity) topology. The Alexandroff compactification of the $ MW $-topological plane is called the infinite $ MW $-topological sphere up to homeomorphism. We first prove that under the $ MW $-topology on $ {\mathbb Z}^2 $ the connectedness of $ X(\subset {\mathbb Z}^2) $ with $ X^\sharp\geq 2 $ implies the semi-openness of $ X $. Besides, for the infinite $ MW $-topological sphere, we introduce a new condition for the hereditary property of the compactness of it. In addition, we investigate some conditions preserving the semi-openness or semi-closedness of a subset of the $ MW $-topological plane in the process of an Alexandroff compactification. Finally, we prove that the infinite $ MW $-topological sphere is a semi-regular space; thus, it is a semi-$ T_3 $-space because it is a semi-$ T_1 $-space. Hence we finally conclude that an Alexandroff compactification of the $ MW $-topological plane preserves the semi-$ T_3 $ separation axiom.
Citation: Sik Lee, Sang-Eon Han. Semi-separation axioms associated with the Alexandroff compactification of the $ MW $-topological plane[J]. Electronic Research Archive, 2023, 31(8): 4592-4610. doi: 10.3934/era.2023235
The present paper aims to investigate some semi-separation axioms relating to the Alexandroff one point compactification (Alexandroff compactification, for short) of the digital plane with the Marcus-Wyse ($ MW $-, for brevity) topology. The Alexandroff compactification of the $ MW $-topological plane is called the infinite $ MW $-topological sphere up to homeomorphism. We first prove that under the $ MW $-topology on $ {\mathbb Z}^2 $ the connectedness of $ X(\subset {\mathbb Z}^2) $ with $ X^\sharp\geq 2 $ implies the semi-openness of $ X $. Besides, for the infinite $ MW $-topological sphere, we introduce a new condition for the hereditary property of the compactness of it. In addition, we investigate some conditions preserving the semi-openness or semi-closedness of a subset of the $ MW $-topological plane in the process of an Alexandroff compactification. Finally, we prove that the infinite $ MW $-topological sphere is a semi-regular space; thus, it is a semi-$ T_3 $-space because it is a semi-$ T_1 $-space. Hence we finally conclude that an Alexandroff compactification of the $ MW $-topological plane preserves the semi-$ T_3 $ separation axiom.
| [1] | J. R. Munkres, Topology: A First Course, Pearson College Div, 1974. |
| [2] | P. Alexandorff, Uber die Metrisation der im Kleinen kompakten topologischen Räume, Math. Ann., 92 (1924), 294–301. |
| [3] |
N. Levine, Semi-open sets and semi-continuity in topological spaces, Mathematics, 70 (1963), 36–41. https://doi.org/10.1080/00029890.1963.11990039 doi: 10.1080/00029890.1963.11990039
|
| [4] | C. Dorsett, Semi-regular spaces, Soochow J. Math., 8 (1982), 45–53. |
| [5] | S. N. Maheshwari, R. Prasad, Some new separation axioms, Ann. Soc. Sci. Bruxelles 89 (1975), 395–402. |
| [6] | S. N. Maheshwari, R. Prasad, On $s$-regular spaces, Glasnik Math., 10 (1975), 347–350. |
| [7] | T. Noiri, A note on $s$-regular space, Glasnik Math., 13 (1978), 107–110. |
| [8] | P. Bhattacharyya, B. K. Lahiri, Semi-generalized closed set in topology, Indian J. Math., 29 (1987), 375–382. |
| [9] | N. Biswas, On some mappings in topological spaces, Ph.D thesis, University of Calcutta, 1971. |
| [10] |
S. G. Crosseley, A note on semitopological properties, Proc. Amer. Math. Soc., 72 (1978), 409–412. https://doi.org/10.1090/S0002-9939-1978-0507348-9 doi: 10.1090/S0002-9939-1978-0507348-9
|
| [11] | S. G. Crosseley, S. K. Hidelfrand, Semi-closure, Texas. J. Sci., 22 (1971), 99–112. |
| [12] |
S. G. Crosseley, S. K. Hidelfrand, Semi-topological properties, Fund. Math., 74 (1972), 233–254. https://doi.org/10.4064/fm-74-3-233-254 doi: 10.4064/fm-74-3-233-254
|
| [13] | M. C. Cueva, R. K. Saraf, A research on characterizations of semi-$T_{\frac 1{2}}$ spaces, Divulg. Math., 8 (2000), 43–50. |
| [14] | C. Dorsett, Semi-separtaion axioms and hyperspaces, Int. J. Math. Sci., 4 (1981), 445–450. |
| [15] | N. Hamllet, A correction to the paper "Semi-open sets and semi-continuity in topological spaces" by Norman Levine, Proc. Amer. Math. Soc., 49 (1975), 458–460. |
| [16] | S. E. Han, Study on topological spaces with the semi-$T_{\frac{1}{2}}$-separation axiom, Honam Math. Jour., 35 (2013), 717–726. |
| [17] |
S. E. Han, Hereditary properties of semi-separation axioms and their applications, Filomat, 32 (2018), 4689–4700. https://doi.org/10.2298/FIL1813689H doi: 10.2298/FIL1813689H
|
| [18] |
S. E. Han, Topologies of the quotient spaces induced by the $M$-topological plane and the infinite $M$-topological sphere, Topol. Appl., 264 (2019), 201–209. https://doi.org/10.1016/j.topol.2019.06.016 doi: 10.1016/j.topol.2019.06.016
|
| [19] | S. E. Han, Semi-topological properties of the Marcus-Wyse topoloigical spaces, AIMS Math., 7 (2022), 12742–12759. |
| [20] |
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
|
| [21] | D. Jankovic, I. Reilly, On semi-separation axioms, Indian J. Pure Appl. Math., 16 (1985), 957–964. |
| [22] |
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
|
| [23] | J. P. Penot et M. Thera, Semi-continuous mappings in general topology, Arch. Math., 38 (1982), 158–166. |
| [24] | V. A. Chatyrko, S. E. Han, Y. Hattori, Some Remarks Concerning Semi-$T_{\frac{1}{2}}$- spaces, Filomat, 28 (2014), 21–25. |
| [25] |
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
|
| [26] |
S. E. Han, Semi-Jordan curve theorem on the Marcus-Wyse topological plane, Electron. Res. Arch., 30 (2022), 4341–4365. https://doi.org/10.3934/era.2022220 doi: 10.3934/era.2022220
|
| [27] | F. Wyse, D. Marcus, , A special topology for the integers (problem 5712), Amer. Math., 77 (1970), 1119. |
| [28] |
S. E. Han, S. Özçaǧ, The fixed point property of the infinite $M$-sphere, Mathematics, 8 (2020). https://doi.org/10.3390/math8040599 doi: 10.3390/math8040599
|
| [29] | P. Alexandorff, Diskrete Räume, Mat. Sb., 2 (1937), 501–519. |
| [30] |
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
|
| [31] |
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
|
| [32] |
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
|
| [33] | T. Y. Kong, A. Rosenfeld, Topological Algorithms for the Digital Image Processing, Elsevier Science, Amsterdam, 1996. |
| [34] | J. ${\check S}$lapal, Digital Jordan curves, Topol. Appl., 153 (2006), 3255-3264. |
| [35] | S. E. Han, Topological graphs based on a new topology on ${\mathbb Z}^n$ and its applications, Filomat, 31 (2017), 6313–6328. |
| [36] | W. Dunham, $T_{\frac{1}{2}}$-spaces, Kyungpook Math. J., 17 (1977), 161–169. |
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Sik Lee, Sang-Eon Han. Semi-separation axioms associated with the Alexandroff compactification of the $ MW $-topological plane[J]. Electronic Research Archive, 2023, 31(8): 4592-4610. doi: 10.3934/era.2023235
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