# Article

 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 .

## Files

Files Size Format View
MathBohem_141-2016-4_4.pdf 250.3Kb application/pdf View/Open

Partner of