Three new soft separation axioms in soft topological spaces

Dina Abuzaid; Samer Al Ghour; Dina Abuzaid; Samer Al Ghour

doi:10.3934/math.2024223

AIMS Mathematics

2024, Volume 9, Issue 2: 4632-4648. doi: 10.3934/math.2024223

Previous Article Next Article

Research article

Three new soft separation axioms in soft topological spaces

Dina Abuzaid ¹,
Samer Al Ghour ^{2
,
,}

1.
Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
2.
Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid 22110, Jordan

Received: 11 December 2023 Revised: 08 January 2024 Accepted: 12 January 2024 Published: 18 January 2024
MSC : 54A40, 54D05

Soft $ \omega $-almost-regularity, soft $ \omega $ -semi-regularity, and soft $ \omega $-$ T_{2\frac{1}{2}} $ as three novel soft separation axioms are introduced. It is demonstrated that soft $ \omega $ -almost-regularity is strictly between "soft regularity" and "soft almost-regularity"; soft $ \omega $-$ T_{2\frac{1}{2}} $ is strictly between "soft $ T_{2\frac{1}{2}} $" and "soft $ T_{2} $", and soft $ \omega $ -semi-regularity is a weaker form of both "soft semi-regularity" and "soft $ \omega $-regularity". Several sufficient conditions for the equivalence between these new three notions and some of their relevant ones are given. Many characterizations of soft $ \omega $-almost-regularity are obtained, and a decomposition theorem of soft regularity by means of "soft $ \omega $ -semi-regularity" and "soft $ \omega $-almost-regularity" is obtained. Furthermore, it is shown that soft $ \omega $-almost-regularity is heritable for specific kinds of soft subspaces. It is also proved that the soft product of two soft $ \omega $-almost regular soft topological spaces is soft $ \omega $-almost regular. In addition, the connections between our three new conceptions and their topological counterpart topological spaces are discussed.
- soft almost regularity,
- soft $ \omega $ -openness,
- soft $ R\omega $-openness,
- soft regularity,
- soft $ \omega $ -regularity,
- soft semi-regularity
Citation: Dina Abuzaid, Samer Al Ghour. Three new soft separation axioms in soft topological spaces[J]. AIMS Mathematics, 2024, 9(2): 4632-4648. doi: 10.3934/math.2024223

Related Papers:

Abstract

Soft $ \omega $-almost-regularity, soft $ \omega $ -semi-regularity, and soft $ \omega $-$ T_{2\frac{1}{2}} $ as three novel soft separation axioms are introduced. It is demonstrated that soft $ \omega $ -almost-regularity is strictly between "soft regularity" and "soft almost-regularity"; soft $ \omega $-$ T_{2\frac{1}{2}} $ is strictly between "soft $ T_{2\frac{1}{2}} $" and "soft $ T_{2} $", and soft $ \omega $ -semi-regularity is a weaker form of both "soft semi-regularity" and "soft $ \omega $-regularity". Several sufficient conditions for the equivalence between these new three notions and some of their relevant ones are given. Many characterizations of soft $ \omega $-almost-regularity are obtained, and a decomposition theorem of soft regularity by means of "soft $ \omega $ -semi-regularity" and "soft $ \omega $-almost-regularity" is obtained. Furthermore, it is shown that soft $ \omega $-almost-regularity is heritable for specific kinds of soft subspaces. It is also proved that the soft product of two soft $ \omega $-almost regular soft topological spaces is soft $ \omega $-almost regular. In addition, the connections between our three new conceptions and their topological counterpart topological spaces are discussed.

References

[1]	D. Molodtsov, Soft set theory first results, Comput. Math. Appl., 37 (1999), 19–31. http://doi.org/10.1016/S0898-1221(99)00056-5 doi: 10.1016/S0898-1221(99)00056-5
[2]	P. K. Maji, R. Biswas, A. R. Roy, Soft set theory, Comput. Math. Appl., 45 (2003), 555–562. http://doi.org/10.1016/S0898-1221(03)00016-6 doi: 10.1016/S0898-1221(03)00016-6
[3]	M. I. Ali, F. Feng, X. Liu, W. K. Min, M. Shabir, On some new operations in soft set theory, Comput. Math. Appl., 57 (2009), 1547–1553. https://doi.org/10.1016/j.camwa.2008.11.009 doi: 10.1016/j.camwa.2008.11.009
[4]	K. V. Babitha, J. J. Sunil, Soft set relations and functions, Comput. Math. Appl., 60 (2010), 1840–1849. https://doi.org/10.1016/j.camwa.2010.07.014 doi: 10.1016/j.camwa.2010.07.014
[5]	K. Qin, Z. Hong, On soft equality, J. Computat. Appl. Math., 234 (2010), 1347–1355. https://doi.org/10.1016/j.cam.2010.02.028 doi: 10.1016/j.cam.2010.02.028
[6]	T. M. Al-shami, Investigation and corrigendum to some results related to $g$-soft equality and $gf$-soft equality relations, Filomat, 33 (2019), 3375–3383. https://doi.org/10.2298/FIL1911375A doi: 10.2298/FIL1911375A
[7]	T. M. Al-shami, M. E. El-Shafei, $T$-soft equality relation, Turk. J. Math., 44 (2020), 1427–1441. https://doi.org/10.3906/MAT-2005-117 doi: 10.3906/MAT-2005-117
[8]	M. Ali, H. Khan, L. H. Son, F. Smarandache, W. B. V. Kandasamy, New soft sets based class of linear algebraic codes, Symmetry, 10 (2018), 510. https://doi.org/10.3390/sym10100510 doi: 10.3390/sym10100510
[9]	N. Cagman, S. Enginoglu, Soft matrix theory and its decision making, Comput. Math. Appl., 59 (2010), 3308–3314. https://doi.org/10.1016/j.camwa.2010.03.015 doi: 10.1016/j.camwa.2010.03.015
[10]	F. Karaaslan, Soft classes and soft rough classes with applications in decision making, Math. Probl. Eng., 2016 (2016), 1584528. https://doi.org/10.1155/2016/1584528 doi: 10.1155/2016/1584528
[11]	S. Yuksel, T. Dizman, G. Yildizdan, U. Sert, Application of soft sets to diagnose the prostate cancer risk, J. Inequal. Appl., 2013 (2013), 229. https://doi.org/10.1186/1029-242X-2013-229 doi: 10.1186/1029-242X-2013-229
[12]	M. Shabir, M. Naz, On soft topological spaces, Comput. Math. Appl., 61 (2011), 1786–1799. https://doi.org/10.1016/j.camwa.2011.02.006 doi: 10.1016/j.camwa.2011.02.006
[13]	S. Hussain, B. Ahmad, Soft separation axioms in soft topological spaces, Hacet. J. Math. Stat., 44 (2015), 559–568. https://doi.org/10.15672/HJMS.2015449426 doi: 10.15672/HJMS.2015449426
[14]	M. E. El-Shafei, T. M. Al-shami, Applications of partial belong and total non-belong relations on soft separation axioms and decision-making problem, Comput. Appl. Math., 39 (2020), 138. https://doi.org/10.1007/s40314-020-01161-3 doi: 10.1007/s40314-020-01161-3
[15]	T. M. Al-shami, M. E. El-Shafei, Partial belong relation on soft separation axioms and decision-making problem, two birds with one stone, Soft Comput., 24 (2020), 5377–5387. https://doi.org/10.1007/s00500-019-04295-7 doi: 10.1007/s00500-019-04295-7
[16]	T. M. Al-Shami, On soft separation axioms and their applications on decision-making problem, Math. Probl. Eng., 2021 (2021), 1–12. https://doi.org/10.1155/2021/8876978 doi: 10.1155/2021/8876978
[17]	A. Aygunoglu, H. Aygun, Some notes on soft topological spaces, Neural Comput. Appl., 21 (2012), 113–119. https://doi.org/10.1007/s00521-011-0722-3 doi: 10.1007/s00521-011-0722-3
[18]	E. Peyghan, B. Samadi, A. Tayebi, Some results related to soft topological spaces, Facta Univ.-Ser. Math., 29 (2014), 325–336.
[19]	T. M. Al-Shami, Compactness on soft topological ordered spaces and its application on the information system, J. Math., 2021 (2021), 1–12. https://doi.org/10.1155/2021/6699092 doi: 10.1155/2021/6699092
[20]	S. Al Ghour, Z. A. Ameen, Maximal soft compact and maximal soft connected topologies, Appl. Comput. Intell. S., 2022 (2022), 2060808. https://doi.org/10.1155/2022/9860015 doi: 10.1155/2022/9860015
[21]	H. H. Al-Jarrah, A. Rawshdeh, T. M. Al-shami, On soft compact and soft Lindelöf spaces via soft regular closed sets, Afr. Mat., 33 (2022), 23. https://doi.org/10.1007/s13370-021-00952-z doi: 10.1007/s13370-021-00952-z
[22]	T. M. Al-shami, A. Mhemdi, R. Abu-Gdairi, M. E. El-Shafei, Compactness and connectedness via the class of soft somewhat open sets, AIMS Math., 8 (2022), 815–840. https://doi.org/10.3934/math.2023040 doi: 10.3934/math.2023040
[23]	S. Hussain, A note on soft connectedness, J. Egypt. Math. Soc., 23 (2015), 6–11. https://doi.org/10.1016/j.joems.2014.02.003 doi: 10.1016/j.joems.2014.02.003
[24]	H. L. Yang, X. Liao, S. G. Li, On soft continuous mappings and soft connectedness of soft topological spaces, Hacet. J. Math. Stat., 44 (2015), 385–398. https://doi.org/10.15672/HJMS.2015459876 doi: 10.15672/HJMS.2015459876
[25]	T. M. Al-Shami, , E. S. A. Abo-Tabl, Connectedness and local connectedness on infra soft topological spaces, Mathematics, 9 (2021), 1759. https://doi.org/10.3390/math9151759 doi: 10.3390/math9151759
[26]	S. S. Thakur, A. S. Rajput, Connectedness between soft sets, New Math. Nat. Comput., 14 (2018), 53–71. https://doi.org/10.1142/S1793005718500059 doi: 10.1142/S1793005718500059
[27]	T. M. Al-shami, L. D. R. Kocinac, The equivalence between the enriched and extended soft topologies, Appl. Comput. Math., 18 (2019), 149–162.
[28]	M. Terepeta, On separating axioms and similarity of soft topological spaces, Soft Comput., 23 (2019), 1049–1057. https://doi.org/10.1007/s00500-017-2824-z doi: 10.1007/s00500-017-2824-z
[29]	O. Tantawy, S. A. El-Sheikh, S. Hamde, Separation axioms on soft topological spaces, Ann. Fuzzy Math. Inform., 11 (2016), 511–525.
[30]	S. Bayramov, C. G. Aras, A new approach to separability and compactness in soft topological spaces, TWMS J. Pure Appl. Math., 9 (2018), 82–93.
[31]	A. K. Prasad, S. S. Thakur, Soft almost regular spaces, Malaya J. Mat., 7 (2019), 408–411. https://doi.org/10.26637/MJM0703/0007 doi: 10.26637/MJM0703/0007
[32]	S. Ramkumar, V. Subbiah, Soft separation axioms and soft product of soft topological spaces, Maltepe J. Math., 2 (2020), 61–75.
[33]	S. Al Ghour, Weaker forms of soft regular and soft $T_{2}$ soft topological spaces, Mathematics, 9 (2021), 2153. https://doi.org/10.3390/math9172153 doi: 10.3390/math9172153
[34]	S. Al Ghour, A. Bin-Saadon, On some generated soft topological spaces and soft homogeneity, Heliyon, 5 (2019). https://doi.org/10.1016/j.heliyon.2019.e02061
[35]	S. Al Ghour, W. Hamed, On two classes of soft sets in soft topological spaces, Symmetry, 12 (2020), 265. https://doi.org/10.3390/sym12020265 doi: 10.3390/sym12020265
[36]	S. Al Ghour, Soft $R\omega $-open sets and the soft topology of soft $\delta _{\omega }$-open sets, Axioms, 11 (2022), 177. https://doi.org/10.3390/axioms11040177 doi: 10.3390/axioms11040177
[37]	M. K. Singal, S. P. Arya, On almost regular spaces, Glasnik Mat., 4 (1969), 89–99. doi: 10.3390/sym12020265
[38]	N. Levine, Generalized closed sets in topology, Rend. Circ. Mat. Palerm., 19 (1970), 89–96. https://doi.org/10.1007/BF02843888 doi: 10.1007/BF02843888
[39]	S. Al Ghour, On $\omega $-almost-regularity and $\omega $ -semi-regularity in topological spaces, J. Math. Comput. Sci., 31 (2023), 188–196. https://doi.org/10.22436/jmcs.031.02.05 doi: 10.22436/jmcs.031.02.05
[40]	S. Al Ghour, Strong form of soft semi-open sets in soft topological spaces, Int. J. Fuzzy Log. Inte., 21 (2021), 159–168. https://doi.org/10.5391/IJFIS.2021.21.2.159 doi: 10.5391/IJFIS.2021.21.2.159
[41]	L. A. Steen, J. Seebach, Counterexamples in topology, New York: Springer, 1970. https://doi.org/10.1007/978-1-4612-6290-9

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)