Geodesic
$R_z$-supercontinuous functions
Mathematica Bohemica, Tome 140 (2015) no. 3, pp. 329-343

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MR Zbl
A new class of functions called “$R_{z}$-supercontinuous functions” is introduced. Their basic properties are studied and their place in the hierarchy of strong variants of continuity that already exist in the literature is elaborated. The class of $R_{z}$-supercontinuous functions properly includes the class of $R_{\rm cl}$-supercontinuous functions, Tyagi, Kohli, Singh (2013), which in its turn contains the class of $\rm cl$-supercontinuous ($\equiv $ clopen continuous) functions, Singh (2007), Reilly, Vamanamurthy (1983), and is strictly contained in the class of $R_{\delta }$-supercontinuous, Kohli, Tyagi, Singh, Aggarwal (2014), which in its turn is properly contained in the class of $R$-supercontinuous functions, Kohli, Singh, Aggarwal (2010).
A new class of functions called “$R_{z}$-supercontinuous functions” is introduced. Their basic properties are studied and their place in the hierarchy of strong variants of continuity that already exist in the literature is elaborated. The class of $R_{z}$-supercontinuous functions properly includes the class of $R_{\rm cl}$-supercontinuous functions, Tyagi, Kohli, Singh (2013), which in its turn contains the class of $\rm cl$-supercontinuous ($\equiv $ clopen continuous) functions, Singh (2007), Reilly, Vamanamurthy (1983), and is strictly contained in the class of $R_{\delta }$-supercontinuous, Kohli, Tyagi, Singh, Aggarwal (2014), which in its turn is properly contained in the class of $R$-supercontinuous functions, Kohli, Singh, Aggarwal (2010).
DOI : 10.21136/MB.2015.144399
Classification : 54C08, 54C10
Keywords: $z$-supercontinuous function; $F$-supercontinuous function; $\rm cl$-supercontinuous function; $R_z$-supercontinuous function; $R$-supercontinuous function; $r_z$-open set; $r_z$-closed set; $z$-embedded set; $R_z$-space; functionally Hausdorff space 
Singh, Davinder; Tyagi, Brij Kishore; Aggarwal, Jeetendra; Kohli, Jogendra K. $R_z$-supercontinuous functions. Mathematica Bohemica, Tome 140 (2015) no. 3, pp. 329-343. doi: 10.21136/MB.2015.144399
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[1] Alò, R. A., Shapiro, H. L.: Normal Topological Spaces. Cambridge Tracts in Mathematics 65 Cambridge University Press, Cambridge (1974). | MR | Zbl

[2] Aull, C. E.: Functionally regular spaces. Nederl. Akad. Wet., Proc., Indag. Math. 38, Ser. A 79 (1976), 281-288. | DOI | MR | Zbl

[3] Beckhoff, F.: Topologies on the ideal space of a {B}anach algebra and spectral synthesis. Proc. Am. Math. Soc. 125 (1997), 2859-2866. | DOI | MR | Zbl

[4] Beckhoff, F.: Topologies of compact families on the ideal space of a {B}anach algebra. Stud. Math. 118 (1996), 63-75. | DOI | MR | Zbl

[5] Beckhoff, F.: Topologies on the space of ideals of a {B}anach algebra. Stud. Math. 115 (1995), 189-205. | DOI | MR | Zbl

[6] Blair, R. L., Hager, A. W.: Extensions of zero-sets and of real-valued functions. Math. Z. 136 (1974), 41-52. | DOI | MR | Zbl

[7] Gauld, D. B., Mršević, M., Reilly, I. L., Vamanamurthy, M. K.: Continuity properties of functions. Topology, Theory and Applications, ed. Á. Császár, 5th Colloq., Eger, Hungary, 1983, Colloq. Math. Soc. János Bolyai 41 North-Holland, Amsterdam; János Bolyai Mathematical Society, Budapest (1985), 311-322. | MR | Zbl

[8] Gleason, A. M.: Universal locally connected refinements. Ill. J. Math. 7 (1963), 521-531. | DOI | MR | Zbl

[9] Kohli, J. K.: Change of topology, characterizations and product theorems for semilocally {$P$}-spaces. Houston J. Math. 17 (1991), 335-350. | MR | Zbl

[10] Kohli, J. K.: A framework including the theories of continuous functions and certain noncontinuous functions. Note Mat. 10 (1990), 37-45. | MR

[11] Kohli, J. K.: A unified approach to continuous and certain noncontinuous functions. I. J. Aust. Math. Soc., Ser. A 48 (1990), 347-358. | DOI | MR

[12] Kohli, J. K.: A unified approach to continuous and certain noncontinuous functions. {II}. Bull. Aust. Math. Soc. 41 (1990), 57-74. | DOI | MR

[13] Kohli, J. K.: A unified view of (complete) regularity and certain variants of (complete) regularity. Can. J. Math. 36 (1984), 783-794. | DOI | MR | Zbl

[14] Kohli, J. K.: A class of mappings containing all continuous and all semiconnected mappings. Proc. Am. Math. Soc. 72 (1978), 175-181. | DOI | MR | Zbl

[15] Kohli, J. K., Kumar, R.: $z$-supercontinuous functions. Indian J. Pure Appl. Math. 33 (2002), 1097-1108. | MR | Zbl

[16] Kohli, J. K., Singh, D.: $D_\delta$-supercontinuous functions. Indian J. Pure Appl. Math. 34 (2003), 1089-1100. | MR | Zbl

[17] Kohli, J. K., Singh, D.: $D$-supercontinuous functions. Indian J. Pure Appl. Math. 32 (2001), 227-235. | MR | Zbl

[18] Kohli, J. K., Singh, D., Aggarwal, J.: $R$-supercontinuous functions. Demonstr. Math. (electronic only) 43 (2010), 703-723. | MR | Zbl

[19] Kohli, J. K., Singh, D., Aggarwal, J.: $F$-supercontinuous functions. Appl. Gen. Topol. (electronic only) 10 (2009), 69-83. | DOI | MR | Zbl

[20] Kohli, J. K., Singh, D., Kumar, R.: Some properties of strongly {$\theta$}-continuous functions. Bull. Calcutta Math. Soc. 100 (2008), 185-196. | MR

[21] Kohli, J. K., Tyagi, B. K., Singh, D., Aggarwal, J.: $R_\delta$-supercontinuous functions. Demonstr. Math. (electronic only) 47 (2014), 433-448. | MR | Zbl

[22] Levine, N.: Strong, continuity in topological spaces. Am. Math. Mon. 67 (1960), 269. | DOI | Zbl

[23] Long, P. E., Herrington, L. L.: Strongly $\theta $-continuous functions. J. Korean Math. Soc. 18 (1981), 21-28. | MR | Zbl

[24] Mack, J.: Countable paracompactness and weak normality properties. Trans. Am. Math. Soc. 148 (1970), 265-272. | DOI | MR | Zbl

[25] Munshi, B. M., Bassan, D. S.: Super-continuous mappings. Indian J. Pure Appl. Math. 13 (1982), 229-236. | MR | Zbl

[26] Noiri, T.: Supercontinuity and some strong forms of continuity. Indian J. Pure Appl. Math. 15 (1984), 241-250. | MR

[27] Noiri, T.: On $\delta $-continuous functions. J. Korean Math. Soc. 16 (1980), 161-166. | MR | Zbl

[28] Reilly, I. L., Vamanamurthy, M. K.: On super-continuous mappings. Indian J. Pure Appl. Math. 14 (1983), 767-772. | MR | Zbl

[29] Singal, M. K., Nimse, S. B.: $z$-continuous mappings. Math. Stud. 66 (1997), 193-210. | MR | Zbl

[30] Singh, D.: $\rm cl$-supercontinuous functions. Appl. Gen. Topol. 8 (2007), 293-300. | DOI | MR

[31] Singh, D.: $D^*$-supercontinuous functions. Bull. Calcutta Math. Soc. 94 (2002), 67-76. | MR | Zbl

[32] Singh, D., Kohli, J. K.: Separation axioms between functionally regular spaces and $R_{0}$-spaces. Submitted to Sci. Stud. Res., Ser. Math. Inform.

[33] Somerset, D. W. B.: Ideal spaces of Banach algebras. Proc. Lond. Math. Soc. (3) 78 (1999), 369-400. | DOI | MR | Zbl

[34] L. A. Steen, J. A. Seebach, Jr.: Counterexamples in Topology. Springer, New York (1978). | MR | Zbl

[35] Tyagi, B. K., Kohli, J. K., Singh, D.: $R_ cl$-supercontinuous functions. Demonstr. Math. (electronic only) 46 (2013), 229-244. | MR | Zbl

[36] Est, W. T. van, Freudenthal, H.: Trennung durch stetige Funktionen in topologischen Räumen. Nederl. Akad. Wet., Proc., Indagationes Math. 13, Ser. A 54 German (1951), 359-368. | DOI | MR

[37] Veličko, N. V.: $H$-closed topological spaces. Transl., Ser. 2, Am. Math. Soc. 78 (1968), 103-118 translation from Russian original, Mat. Sb. (N.S.), 70 98-112 (1966). | MR

[38] G. S. Young, Jr.: The introduction of local connectivity by change of topology. Am. J. Math. 68 (1946), 479-494. | DOI | MR | Zbl

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