In this paper, we define an almost $ \delta b $-continuity, which is a weaker form of $ R $-map and we investigate and obtain its some properties and characterizations. Finally, we show that a function $ f:\left(X, \tau \right) \rightarrow \left(Y, \varphi \right) $ is almost $ \delta b $-continuous if and only if $ f:\left(X, \tau _{s}\right) \rightarrow \left(Y, \varphi _{s}\right) $ is $ b $-continuous, where $ \tau _{s} $ and $ \varphi _{s} $ are semiregularizations of $ \tau $ and $ \varphi $, respectively.
Citation: Cenap Ozel, M. A. Al Shumrani, Aynur Keskin Kaymakci, Choonkil Park, Dong Yun Shin. On $ \delta b $-open continuous functions[J]. AIMS Mathematics, 2021, 6(3): 2947-2955. doi: 10.3934/math.2021178
In this paper, we define an almost $ \delta b $-continuity, which is a weaker form of $ R $-map and we investigate and obtain its some properties and characterizations. Finally, we show that a function $ f:\left(X, \tau \right) \rightarrow \left(Y, \varphi \right) $ is almost $ \delta b $-continuous if and only if $ f:\left(X, \tau _{s}\right) \rightarrow \left(Y, \varphi _{s}\right) $ is $ b $-continuous, where $ \tau _{s} $ and $ \varphi _{s} $ are semiregularizations of $ \tau $ and $ \varphi $, respectively.
| [1] | D. Andrijević, On $b$-open sets, Mat. Vesnik, 48 (1996), 59–64. |
| [2] | D. A. Carnahan, Some properties related to compactness in topological spaces, Thesis (Ph.D.), University of Arkansas, 1973. |
| [3] | E. Ekici, On $\delta$-semiopen sets and a generalization of functions, Bol. Soc. Parana. Mat., 23 (2005), 73–84. |
| [4] |
E. Ekici, Generalization of perfectly continuous, regular set connected and clopen functions, Acta Math. Hung., 107 (2005), 193–206. doi: 10.1007/s10474-005-0190-2
|
| [5] | A. El-Atik, A study on some types of mappings on topological spaces, M. Sc. Thesis, Tanta University, Egypt, 1997. |
| [6] | A. I. El-Magharabi, A. M. Mubarki, $Z$-open sets and $Z$-continuity in topological space, Int. J. Math. Arch., 2 (2011), 1819–1827. |
| [7] |
L. L. Herrington, Properties of nearly-compact spaces, Proc. Am. Math. Soc., 45 (1974), 431–436. doi: 10.1090/S0002-9939-1974-0346748-3
|
| [8] | A. K. Kaymakci, A new class of generalized $b$-open sets, IV. INSAC International Natural and Engineering Sciences Congress, 2019. |
| [9] | A. K. Kaymakci, Weakly $\delta$-$b$-continuous functions, Gen. Math. Notes, 27 (2015), 24–39. |
| [10] |
A. Keskin, T. Noiri, Almost $b$-continuous functions, Chaos Solitons Fractals, 41 (2009), 72–81. doi: 10.1016/j.chaos.2007.11.012
|
| [11] |
N. Levine, Semi-open sets and semi-continuity in topological spaces, Am. Math. Monthly, 70 (1963), 36–41. doi: 10.1080/00029890.1963.11990039
|
| [12] | B. M. Munshi, D. S. Bassan, Almost semi-continuous mappings, Math. Student, 49 (1981), 229–236. |
| [13] | B. M. Munshi, D. S. Bassan, Super continuous mappings, Indian J. Pure Appl. Math., 13 (1982), 229–236. |
| [14] |
T. Noiri, Regular-closed functions and Hausdorff spaces, Math. Nachr., 99 (1980), 217–219. doi: 10.1002/mana.19800990123
|
| [15] | T. Noiri, Remarks on $\delta$-semi-open sets and $ \delta$-preopen sets, Demonstratio Math., 36 (2003), 1007–1020. |
| [16] | J. Park, B. Lee, M. Son, On $\delta$-semiopen sets in topological spaces, J. Indian Acad. Math., 19 (1997), 59–67. |
| [17] | S. Raychaudhuri, M. N. Mukherje, On $\delta$ -almost continuity and $\delta$-preopen sets, Bull. Inst. Math. Acad. Sinica, 21 (1993), 357–366. |
| [18] |
M. H. Stone, Applications of the theory of Boolean rings to general topology, Trans. Am. Math. Soc., 41 (1937), 375–481. doi: 10.1090/S0002-9947-1937-1501905-7
|
| [19] | N. V. Velićko, $H$-Closed topological spaces, Mat. Sb., 70 (1966), 98–112. |
Cenap Ozel, M. A. Al Shumrani, Aynur Keskin Kaymakci, Choonkil Park, Dong Yun Shin. On $ \delta b $-open continuous functions[J]. AIMS Mathematics, 2021, 6(3): 2947-2955. doi: 10.3934/math.2021178
DownLoad: