| Title:
|
Almost $\tilde g_\alpha$-closed functions and separation axioms (English) |
| Author:
|
Ravi, O. |
| Author:
|
Ganesan, S. |
| Author:
|
Latha, R. |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
137 |
| Issue:
|
3 |
| Year:
|
2012 |
| Pages:
|
275-291 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We introduce a new class of functions called almost $\tilde {g}_{\alpha }$-closed and use the functions to improve several preservation theorems of normality and regularity and also their generalizations. The main result of the paper is that normality and weak normality are preserved under almost $\tilde {g}_{\alpha }$-closed continuous surjections. (English) |
| Keyword:
|
topological space |
| Keyword:
|
$\tilde {g}$-closed set |
| Keyword:
|
$\tilde {g}_{\alpha }$-closed set |
| Keyword:
|
$\alpha g$-closed set |
| MSC:
|
54C05 |
| MSC:
|
54C08 |
| MSC:
|
54C10 |
| MSC:
|
54D15 |
| idZBL:
|
Zbl 1265.54087 |
| idMR:
|
MR3112488 |
| DOI:
|
10.21136/MB.2012.142895 |
| . |
| Date available:
|
2012-08-19T21:21:38Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142895 |
| . |
| Reference:
|
[1] Andrijevic, D.: Some properties of the topology of $\alpha$-sets.Mat. Vesn. 36 (1984), 1-10. Zbl 0546.54003, MR 0880637 |
| Reference:
|
[2] Carnahan, D.: Some properties related to compactness in topological spaces.Ph.D. Thesis, Univ. of Arkansas (1973). MR 2623205 |
| Reference:
|
[3] Devi, R., Balachandran, K., Maki, H.: On generalized $\alpha$-continuous maps and $\alpha$-generalized continuous maps.Far East J. Math. Sci. (1997), 1-15. |
| Reference:
|
[4] Frolík, Z.: Remarks concerning the invariance of Baire spaces under mappings.Czech. Math. J. 11 (1961), 381-385. MR 0133098 |
| Reference:
|
[5] Ganster, M.: On strongly $s$-regular spaces.Glas. Mat., III. Ser. 25 (1990), 195-201. Zbl 0733.54012, MR 1108963 |
| Reference:
|
[6] Greenwood, S., Reilly, I. L.: On feebly closed mappings.Indian J. Pure Appl. Math. 17 (1986), 1101-1105. Zbl 0604.54012, MR 0864149 |
| Reference:
|
[7] Jafari, S., Noiri, T., Rajesh, N., Thivagar, M. L.: Another generalization of closed sets.Kochi J. Math. 3 (2008), 25-38. Zbl 1148.54304, MR 2408589 |
| Reference:
|
[8] Jafari, S., Thivagar, M. L., Paul, Nirmala Rebecca: Remarks on $\tilde{g}_{\alpha}$-closed sets in topological spaces.Int. Math. Forum 5 (2010), 1167-1178. Zbl 1207.54030, MR 2652960 |
| Reference:
|
[9] Jankovic, D. S., Konstadilaki-Savvopoulou, Ch.: On $\alpha$-continuous functions.Math. Bohem. 117 (1992), 259-270. Zbl 0802.54005, MR 1184539 |
| Reference:
|
[10] Levine, N.: Generalized closed sets in topology.Rend. Circ. Mat. Palermo, II. Ser. 19 (1970), 89-96. Zbl 0231.54001, MR 0305341, 10.1007/BF02843888 |
| Reference:
|
[11] Levine, N.: Semi-open sets and semi-continuity in topological spaces.Am. Math. Mon. 70 (1963), 36-41. Zbl 0113.16304, MR 0166752, 10.2307/2312781 |
| Reference:
|
[12] Long, P. E., Herrington, L. L.: Basic properties of regular-closed functions.Rend Circ. Mat. Palermo, II. Ser. 27 (1978), 20-28. Zbl 0416.54005, MR 0542230 |
| Reference:
|
[13] Maki, H., Devi, R., Balachandran, K.: Generalized $\alpha$-closed sets in topology.Bull. Fukuoka Univ. Educ., Part III 42 (1993), 13-21. Zbl 0888.54005 |
| Reference:
|
[14] Maki, H., Devi, R., Balachandran, K.: Associated topologies of generalized $\alpha$-closed sets and $\alpha$-generalized closed sets.Mem. Fac. Sci., Kochi Univ., Ser. A 15 (1994), 51-63. Zbl 0821.54002, MR 1262966 |
| Reference:
|
[15] Maki, H., Rao, K. Chandrasekhara, Gani, A. Nagoor: On generalizing semi-open and preopen sets.Pure Appl. Math. Sci. 49 (1999), 17-29. MR 1696955 |
| Reference:
|
[16] Malghan, S. R.: Generalized closed maps.J. Karnatak Univ., Sci. 27 (1982), 82-88. Zbl 0578.54008, MR 0773568 |
| Reference:
|
[17] Mashhour, A. S., Hasanein, I. A., El-Deeb, S. N.: $\alpha$-continuous and $\alpha$-open mappings.Acta Math. Hung. 41 (1983), 213-218. Zbl 0534.54006, MR 0703734, 10.1007/BF01961309 |
| Reference:
|
[18] Min, W. K.: $\alpha m$-open sets and $\alpha M$-continuous functions.Commun. Korean Math. Soc. 25 (2010), 251-256. Zbl 1211.54030, MR 2662974, 10.4134/CKMS.2010.25.2.251 |
| Reference:
|
[19] Min, W. K., Kim, Y. K.: On weak $M$-semicontinuity on spaces with minimal structures.J. Chungcheong Math. Soc. 23 (2010), 223-229. |
| Reference:
|
[20] Njastad, O.: On some classes of nearly open sets.Pac. J. Math. 15 (1965), 961-970. Zbl 0137.41903, MR 0195040, 10.2140/pjm.1965.15.961 |
| Reference:
|
[21] Noiri, T.: Almost-closed images of countably paracompact spaces.Commentat. Math. 20 (1978), 423-426. Zbl 0398.54007, MR 0519378 |
| Reference:
|
[22] Noiri, T.: Mildly normal spaces and some functions.Kyungpook Math. J. 36 (1996), 183-190. Zbl 0873.54016, MR 1396023 |
| Reference:
|
[23] Noiri, T.: Almost $\alpha g$-closed functions and separation axioms.Acta Math. Hung. 82 (1999), 193-205. Zbl 0924.54020, MR 1674100, 10.1023/A:1026404730639 |
| Reference:
|
[24] Noiri, T., Popa, V.: A unified theory of closed functions.Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 49 (2006), 371-382. Zbl 1119.54304, MR 2281517 |
| Reference:
|
[25] Palaniappan, N., Rao, K. C.: Regular generalized closed sets.Kyungpook Math. J. 33 (1993), 211-219. Zbl 0794.54002, MR 1253673 |
| Reference:
|
[26] Popa, V., Noiri, T.: On $M$-continuous functions.Anal. Univ. ``Dunarea de Jos'', Galati, Ser. Mat. Fiz. Mecan. Teor. Fasc. II. 18 (2000), 31-41. MR 2314773 |
| Reference:
|
[27] Popa, V., Noiri, T.: On the definitions of some generalized forms of continuity under minimal conditions.Mem. Fac. Sci., Kochi Univ., Ser. A 22 (2001), 31-41. Zbl 0972.54011, MR 1822060 |
| Reference:
|
[28] Porter, J. R., Woods, R. G.: Extensions and Absolutes of Hausdorff spaces.Springer, New York (1988). Zbl 0652.54016, MR 0918341 |
| Reference:
|
[29] Ravi, O., Ganesan, S., Chandrasekar, S.: Almost $\alpha gs$-closed functions and separation axioms.Bulletin of Mathematical Analysis and Applications 3 (2011), 165-177. MR 2792611 |
| Reference:
|
[30] Rosas, E., Rajesh, N., Carpintero, C.: Some new types of open and closed sets in minimal structures. II.Int. Math. Forum 4 (2009), 2185-2198. Zbl 1191.54003, MR 2563392 |
| Reference:
|
[31] Singal, M. K., Arya, S. P.: On almost-regular spaces.Glas. Mat., III. Ser. 4 (1969), 89-99. Zbl 0169.24902, MR 0243483 |
| Reference:
|
[32] Singal, M. K., Arya, S. P.: Almost normal and almost completely regular spaces.Glas. Mat., III. Ser. 5 (1970), 141-152. Zbl 0197.18901, MR 0275354 |
| Reference:
|
[33] Singal, M. K., Singal, A. R.: Almost-continuous mappings.Yokohama Math. J. 16 (1968), 63-73. Zbl 0191.20802, MR 0261569 |
| Reference:
|
[34] Singal, M. K., Singal, A. R.: Mildly normal spaces.Kyungpook Math. J. 13 (1973), 27-31. Zbl 0266.54006, MR 0362215 |
| Reference:
|
[35] Kumar, M. K. R. S. Veera: $\hat{g}$-closed sets in topological spaces.Bull. Allahabad Math. Soc. 18 (2003), 99-112. MR 2061436 |
| Reference:
|
[36] Kumar, M. K. R. S. Veera: Between $g^*$-closed sets and $g$-closed sets.Antarct. J. Math. 3 (2006), 43-65. MR 2296082 |
| Reference:
|
[37] Kumar, M. K. R. S. Veera: $^{\sharp} g$-semi-closed sets in topological spaces.Antarct. J. Math. 2 (2005), 201-222. MR 2203685 |
| Reference:
|
[38] Wang, Guojun: On S-closed spaces.Acta Math. Sin. 24 (1981), 55-63. Zbl 0503.54031, MR 0617426 |
| Reference:
|
[39] Yoshimura, M., Miwa, T., Noiri, T.: A generalization of regular closed and $g$-closed functions.Stud. Cercet. Mat. 47 (1995), 353-358. Zbl 0854.54020, MR 1682872 |
| Reference:
|
[40] Zenor, P.: On countable paracompactness and normality.Pr. Mat. 13 (1969), 23-32. Zbl 0242.54016, MR 0248724 |
| . |
