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Keywords:
normal space; (weakly) densely normal space; (weakly) $\theta $-normal space; almost normal space; almost $\beta $-normal space; $\kappa $-normal space; (weakly) $\beta $-normal space
normal space; (weakly) densely normal space; (weakly) $\theta $-normal space; almost normal space; almost $\beta $-normal space; $\kappa $-normal space; (weakly) $\beta $-normal space
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.
References:
[1] Arhangel'skii, A. V.: Relative topological properties and relative topological spaces. Topology Appl. 70 87-99 (1996). DOI 10.1016/0166-8641(95)00086-0 | MR 1397067 | Zbl 0848.54016
[2] Arhangel'skii, A. V., Ludwig, L.: On $\alpha$-normal and $\beta$-normal spaces. Commentat. Math. Univ. Carol. 42 (2001), 507-519. MR 1860239 | Zbl 1053.54030
[3] Das, A. K.: Simultaneous generalizations of regularity and normality. Eur. J. Pure Appl. Math. 4 (2011), 34-41. MR 2770026 | Zbl 1213.54031
[4] Das, A. K.: A note on spaces between normal and $\kappa$-normal spaces. Filomat 27 (2013), 85-88. DOI 10.2298/FIL1301085D | MR 3243902
[5] Das, A. K., Bhat, P.: A class of spaces containing all densely normal spaces. Indian J. Math. 57 (2015), 217-224. MR 3362716 | Zbl 1327.54027
[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
[7] R. F. Dickman, Jr., J. R. Porter: {$\theta $}-perfect and $\theta $-absolutely closed functions. Ill. J. Math. 21 (1977), 42-60. DOI 10.1215/ijm/1256049499 | MR 0428261 | Zbl 0351.54010
[8] Kohli, J. K., Das, A. K.: New normality axioms and decompositions of normality. Glas. Mat. Ser. (3) 37 (2002), 163-173. MR 1918103 | Zbl 1042.54014
[9] Kohli, J. K., Das, A. K.: On functionally $\theta$-normal spaces. Appl. Gen. Topol. 6 (2005), 1-14. DOI 10.4995/agt.2005.1960 | MR 2153423 | Zbl 1077.54011
[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. DOI 10.4995/agt.2006.1926 | MR 2295172 | Zbl 1116.54014
[11] Kohli, J. K., Singh, D.: Weak normality properties and factorizations of normality. Acta Math. Hung. 110 (2006), 67-80. DOI 10.1007/s10474-006-0007-y | MR 2198415 | Zbl 1104.54009
[12] Mack, J.: Countable paracompactness and weak normality properties. Trans. Am. Math. Soc. 148 (1970), 265-272. DOI 10.1090/S0002-9947-1970-0259856-3 | MR 0259856 | Zbl 0209.26904
[13] Murtinová, E.: A $\beta$-normal Tychonoff space which is not normal. Commentat. Math. Univ. Carol. 43 (2002), 159-164. MR 1903315 | Zbl 1090.54016
[14] Singal, M. K., Arya, S. P.: Almost normal and almost completely regular spaces. Glas. Mat. Ser. (3) 5 (25) (1970), 141-152. MR 0275354 | Zbl 0197.18901
[15] Singal, M. K., Singal, A. R.: Mildly normal spaces. Kyungpook Math. J. 13 (1973), 27-31. MR 0362215 | Zbl 0266.54006
[16] Ščepin, E. V.: Real functions, and spaces that are nearly normal. Sibirsk. Mat. Ž. 13 (1972), 1182-1196, 1200 Russian. MR 0326656
[17] L. A. Steen, J. A. Seebach, Jr.: Counterexamples in Topology. Springer, New York (1978). MR 0507446 | Zbl 0386.54001
[18] Veličko, N. V.: {$H$}-closed topological spaces. Mat. Sb. (N.S.) 70 (112) (1966), Russian 98-112. MR 0198418
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