Notes on Strongly Semi-Closed Graph
Authors: Alkhazragy A., Al-Hachami A.K.H., Mayah F. | Published: 23.06.2022 |
Published in issue: #3(102)/2022 | |
DOI: 10.18698/1812-3368-2022-3-17-27 | |
Category: Mathematics and Mechanics | Chapter: Substantial Analysis, Complex and Functional Analysis | |
Keywords: closed graph, semi-closed graph, strongly semi-closed graph |
Abstract
In these papers, we study on mapping with strongly semi-closed graph. We first introduce the notion "strongly semi-closed graph" in analogue to the closed graph. Also, we study some of their basic properties. This study proved that: If Y is extremally s-disconnected semi-T_{2}-space and f : X → Y is a set-s-connected surjection, then G(f) is semi-closed. If Y is Hausdorff space and f is almost semi continuous, then G(f) is strongly semi-closed. If Y is semi-T_{2}-space and f is irresolute, then G(f) is strongly semi-closed. Semi-closed mapping and semi-closed graph are two separate concepts. If G(f) is a semi-closed and f is surjection(onto), then Y is semi-T_{1}-space. If the injective S**-open map X is semi**-connected and G(f) is semi-closed then X is semi-T_{2}-space provided it is T_{1}-space and locally semi-connected. S**-closed mapping and semi-closed graph are two separate concepts. If Y is semi-regular space, then the following are equivalent G(f) is semi-closed and G(f) is strongly semi-closed
Please cite this article as:
Alkhazragy A., Al-Hachami A.K.H., Mayah F. Notes on strongly semi-closed graph. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2022, no. 3 (102), pp. 17--27. DOI: https://doi.org/10.18698/1812-3368-2022-3-17-27
References
[1] Levine N. Semi-open sets and semi-continuity in topological spaces. Am. Math. Mon., 1963, vol. 70, iss. 1, pp. 36--41. DOI: https://doi.org/10.1080/00029890.1963.11990039
[2] Levine N. Generalized closed sets in topology. Rend. Circ. Mat. Palermo, 1970, vol. 19, no. 1, pp. 89--96. DOI: https://doi.org/10.1007/BF02843888
[3] Crossley S.G. Semi-closure. Texas J. Sci., 1971, vol. 22, no. 2-3, pp. 99--112.
[4] Dube K.K., Panwar O.S. Some properties of s-connectedness between sets and set s-connected mappings. Indian J. Pure Appl. Math., 1984, vol. 15, no. 4, pp. 343--354.
[5] Mustafa H.I. On connected functions. Mast. Sc. Thesis. University of Al-Mustansirya, 2001.
[6] Mosa A.L. On some types semi-topological groups. Mast. Sc. Thesis. University of Baghdad, 1998.
[7] Hamlett T.R., Herrington L.L. The closed graph and P-closed graph properties in general topology. AMS, 1981.
[8] Mustafa N.R. On monotone mappings and open mappings. Mast. Sc. Thesis. University of Baghdad, 1992.
[9] Munshi B.M., Bassan D.S. Almost semi-continuous mappings. Math. Student, 1981, vol. 49, pp. 239--248.
[10] Long P.E. Functions with closed graphs. Am. Math. Mon., 1969, vol. 76, no. 8, pp. 930--932. DOI: https://doi.org/10.2307/2317955
[11] Dube K.K., Lee J.-Y., Panwar O.S. A note on semi-closed graph. UIT Rep., 1983, vol. 14, no. 2, pp. 379--383.
[12] Herrington L.L., Long P.E. Characterizations of H-closed spaces. Proc. Am. Math. Soc., 1975, vol. 48, iss. 2, pp. 469--475. DOI: https://doi.org/10.1090/S0002-9939-1975-0365485-3
[13] Gupta A., Kishore K. On functions with strongly closed graph. Int. J. Pure Appl. Math., 2016, vol. 110, no. 2, pp. 383--388.
[14] Dube K.K., Chae G.I., Panwar O.S. On mappings with strongly semi-closed graphs. UIT Rep., 1984, vol. 15, no. 2, pp. 373--389.
[15] Husain T. Topology and maps. Plenum press, 1977.