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A decomposition of continuity. (English) Zbl 0609.54012
This paper introduces the following notions. A subset B of a topological space (X,$${\mathcal T})$$ is defined to be an $${\mathcal A}$$-set if $$B=U-W$$ where U is open and W is regular open. A function f: (X,$${\mathcal T})\to (Y,{\mathcal U})$$ is called $${\mathcal A}$$-continuous if $$f^{-1}(V)$$ is an $${\mathcal A}$$- set in X for each open set V in Y. O. Njåstad [Pac. J. Math. 15, 961-970 (1965; Zbl 0137.419)] defined a subset S of (X,$${\mathcal T})$$ to be an $$\alpha$$-set if $$S\subset int(cl(int S))$$. A. S. Mashhour et al [Acta Math. Hung. 41, 213-218 (1983; Zbl 0534.54006)] defined a function f: (X,$${\mathcal T})\to (Y,{\mathcal U})$$ to be $$\alpha$$-continuous if $$f^{-1}(V)$$ is an $$\alpha$$-set in X for each open set V in Y. The main result of this paper is that f is continuous if and only if it is $$\alpha$$-continuous and $${\mathcal A}$$-continuous.
Reviewer: I.L.Reilly

##### MSC:
 54C05 Continuous maps
##### Keywords:
regular open set; $$\alpha$$-set
Full Text:
##### References:
 [1] J. Dugundji,Topology, Allyn and Bacon (Boston, 1972). [2] S. Fomin, Extensions of topological spaces,Ann. Math.,44 (1943), 471–480. · Zbl 0061.39601 [3] Z. Frolík, Remarks concerning the invariance of Baire spaces under mappings,Czech. Math. J.,11 (86) (1961), 381–385. · Zbl 0104.17204 [4] R. V. Fuller, Relations among continuous and various non-continuous functions,Pacific J. Math.,25 (1968), 495–509. · Zbl 0165.25304 [5] K. R. Gentry and H. B. Hoyle III, Somewhat continuous functions,Czech. Math. J. 21 (96) (1971), 5–12. · Zbl 0222.54010 [6] L. L. Herrington, Some properties preserved by the almost continuous functions,Boll. Un. Math. Ital.,10 (1974), 556–568. · Zbl 0304.54008 [7] T. Husain, Almost continuous mappings,Prace Mat.,10 (1966), 1–7. · Zbl 0138.17601 [8] T. Husain,Topology and Maps, Plenum Press (New York, 1977). · Zbl 0401.54001 [9] S. Kempsty, Sur les fonctions quasicontinues,Fund. Math.,19 (1932), 184–197. · JFM 58.0246.01 [10] N. Levine, A decomposition of continuity in topological spaces,Amer. Math. Monthly,68 (1961), 44–46. · Zbl 0100.18601 [11] N. Levine, Semiopen sets and semicontinuity in topological spaces,Amer. Math. Monthly,70 (1963), 36–41. · Zbl 0113.16304 [12] P. E. Long and E. E. McGhee, Jr., Properties of almost continuous functions,Proc. Amer. Math. Soc.,24 (1970), 175–180. [13] P. E. Long and D. A. Carnahan, Comparing almost-continuous functions,Proc. Amer. Math. Soc.,38 (1973), 413–418. · Zbl 0261.54007 [14] P. E. Long and L. L. Herrington, Properties of almost continuous functions,Boll. Un. Math. Ital.,10 (1974), 336–342. · Zbl 0317.54008 [15] A. S. Mashhour, I. A. Hasanein and S. N. El-Deeb, {$$\alpha$$}-continuous and {$$\alpha$$}-open mappings,Acta Math. Hungar.,41 (1983), 213–218. · Zbl 0534.54006 [16] O. Njåstad, On some classes of nearly open sets,Pacific J. Math.,15 (1965), 961–970. · Zbl 0137.41903 [17] T. Noiri, Semi-continuity and weak-continuity,Czech. Math. J.,31 (106) (1981), 314–321. · Zbl 0483.54006 [18] D. A. Rose, On Levine’s decomposition of continuity,Canad. Math. Bull. 21 (1978), 477–481. · Zbl 0394.54004 [19] M. K. Singal and A. R. Singal, Almost continuous mapping,Yokohama Math. J.,2 (1968), 63–73. · Zbl 0191.20802 [20] J. Stallings, Fixed-point theorems for connectivity maps,Fund. Math.,47 (1959), 249–263. · Zbl 0114.39102
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