| Title:
|
Factorizations of normality via generalizations of $\beta $-normality (English) |
| Author:
|
Das, Ananga Kumar |
| Author:
|
Bhat, Pratibha |
| Author:
|
Gupta, Ria |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
141 |
| Issue:
|
4 |
| Year:
|
2016 |
| Pages:
|
463-473 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The notion of $\beta $-normality was introduced and studied by Arhangel'skii, Ludwig in 2001. Recently, almost $\beta $-normal spaces, which is a simultaneous generalization of $\beta $-normal and almost normal spaces, were introduced by Das, Bhat and Tartir. We introduce a new generalization of normality, namely weak $\beta $-normality, in terms of $\theta $-closed sets, which turns out to be a simultaneous generalization of $\beta $-normality and $\theta $-normality. A space $X$ is said to be weakly $\beta $-normal (w$\beta $-normal$)$ if for every pair of disjoint closed sets $A$ and $B$ out of which, one is $\theta $-closed, there exist open sets $U$ and $V$ such that $\overline {A\cap U}=A$, $\overline {B\cap V}=B$ and $\overline {U}\cap \overline {V}=\emptyset $. It is shown that w$\beta $-normality acts as a tool to provide factorizations of normality. (English) |
| Keyword:
|
normal space |
| Keyword:
|
(weakly) densely normal space |
| Keyword:
|
(weakly) $\theta $-normal space |
| Keyword:
|
almost normal space |
| Keyword:
|
almost $\beta $-normal space |
| Keyword:
|
$\kappa $-normal space |
| Keyword:
|
(weakly) $\beta $-normal space |
| MSC:
|
54D15 |
| idZBL:
|
Zbl 06674856 |
| idMR:
|
MR3576793 |
| DOI:
|
10.21136/MB.2016.0048-15 |
| . |
| Date available:
|
2017-01-03T15:15:00Z |
| Last updated:
|
2020-07-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145953 |
| . |
| Reference:
|
[1] Arhangel'skii, A. V.: Relative topological properties and relative topological spaces.Topology Appl. 70 87-99 (1996). Zbl 0848.54016, MR 1397067, 10.1016/0166-8641(95)00086-0 |
| Reference:
|
[2] Arhangel'skii, A. V., Ludwig, L.: On $\alpha$-normal and $\beta$-normal spaces.Commentat. Math. Univ. Carol. 42 (2001), 507-519. Zbl 1053.54030, MR 1860239 |
| Reference:
|
[3] Das, A. K.: Simultaneous generalizations of regularity and normality.Eur. J. Pure Appl. Math. 4 (2011), 34-41. Zbl 1213.54031, MR 2770026 |
| Reference:
|
[4] Das, A. K.: A note on spaces between normal and $\kappa$-normal spaces.Filomat 27 (2013), 85-88. MR 3243902, 10.2298/FIL1301085D |
| Reference:
|
[5] Das, A. K., Bhat, P.: A class of spaces containing all densely normal spaces.Indian J. Math. 57 (2015), 217-224. Zbl 1327.54027, MR 3362716 |
| Reference:
|
[6] Das, A. K., Bhat, P., Tartir, J. K.: On a simultaneous generalization of $\beta$-normality and almost $\beta$-normality.(to appear) in Filomat. MR 3439956 |
| Reference:
|
[7] R. F. Dickman, Jr., J. R. Porter: {$\theta $}-perfect and $\theta $-absolutely closed functions.Ill. J. Math. 21 (1977), 42-60. Zbl 0351.54010, MR 0428261, 10.1215/ijm/1256049499 |
| Reference:
|
[8] Kohli, J. K., Das, A. K.: New normality axioms and decompositions of normality.Glas. Mat. Ser. (3) 37 (2002), 163-173. Zbl 1042.54014, MR 1918103 |
| Reference:
|
[9] Kohli, J. K., Das, A. K.: On functionally $\theta$-normal spaces.Appl. Gen. Topol. 6 (2005), 1-14. Zbl 1077.54011, MR 2153423, 10.4995/agt.2005.1960 |
| Reference:
|
[10] Kohli, J. K., Das, A. K.: A class of spaces containing all generalized absolutely closed (almost compact) spaces.Appl. Gen. Topol. 7 (2006), 233-244. Zbl 1116.54014, MR 2295172, 10.4995/agt.2006.1926 |
| Reference:
|
[11] Kohli, J. K., Singh, D.: Weak normality properties and factorizations of normality.Acta Math. Hung. 110 (2006), 67-80. Zbl 1104.54009, MR 2198415, 10.1007/s10474-006-0007-y |
| Reference:
|
[12] Mack, J.: Countable paracompactness and weak normality properties.Trans. Am. Math. Soc. 148 (1970), 265-272. Zbl 0209.26904, MR 0259856, 10.1090/S0002-9947-1970-0259856-3 |
| Reference:
|
[13] Murtinová, E.: A $\beta$-normal Tychonoff space which is not normal.Commentat. Math. Univ. Carol. 43 (2002), 159-164. Zbl 1090.54016, MR 1903315 |
| Reference:
|
[14] Singal, M. K., Arya, S. P.: Almost normal and almost completely regular spaces.Glas. Mat. Ser. (3) 5 (25) (1970), 141-152. Zbl 0197.18901, MR 0275354 |
| Reference:
|
[15] Singal, M. K., Singal, A. R.: Mildly normal spaces.Kyungpook Math. J. 13 (1973), 27-31. Zbl 0266.54006, MR 0362215 |
| Reference:
|
[16] Ščepin, E. V.: Real functions, and spaces that are nearly normal.Sibirsk. Mat. Ž. 13 (1972), 1182-1196, 1200 Russian. MR 0326656 |
| Reference:
|
[17] L. A. Steen, J. A. Seebach, Jr.: Counterexamples in Topology.Springer, New York (1978). Zbl 0386.54001, MR 0507446 |
| Reference:
|
[18] Veličko, N. V.: {$H$}-closed topological spaces.Mat. Sb. (N.S.) 70 (112) (1966), Russian 98-112. MR 0198418 |
| . |
