$ S $-$ \delta $-connectedness in $ S $-proximity spaces

Abstract

New types of connectedness in $ S $-proximity spaces, named as an $ S $-$\delta$-connectedness, local $ S $-$ \delta $-connectedness are introduced. Also, their inter-relationships are studied. In an $ S $-proximity space $ (X, \delta_{X}) $, the $ S $-$ \delta $-connectedness of a subset $ U $ of $ X $ with respect to $ \delta_{X} $ may not be same as the $ S $-$ \delta $-connectedness of $ U $ with respect to its subspace proximity $ \delta_{U} $. Further, $ S $-$ \delta $-component and $ S $-$ \delta $-treelike spaces are also defined and a number of results are given.

Keywords

• $ S $-proximity space
• $\delta$-Connected
• $ \delta $-component
• locally $\delta$-Connected
• $ \delta $-treelike

References

1. Brouwer A.E., Treelike Spaces and Related Connected Topological Spaces, Mathematical Centre Tracts, Mathematisch centrum, 75, 1977.
2. Cech E., Topological spaces, Wiley London (1966) fr seminar, Brno, 1936-1939, rev. ed. Z. Frolik, M. Katetov.
3. Dimitrijevi´c R., Kocinac Lj., On connectedness of proximity spaces, Mat. Vesnik, 39(1) (1987), 27-35.
4. Dimitrijevi´c R., Kocinac Lj., On treelike proximity spaces, Mat. Vesnik, 39(3) (1987), 257-261.
5. Efremovic V.A., Infinitesimal spaces, Dokl. Akad. Nauk SSSR, 76 (1951), 341-343 (in Russian).
6. Efremovic V.A., The geometry of proximity I, Mat. Sb., 31 (1952), 189-200 (in Russian).
7. Krishna Murti S.B., A set of axioms for topological algebra, J. Indian Math. Soc., 4 (1940), 116-119.
8. Modak S., Noiri T., A weaker form of connectedness, Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat., 65 (2016), 49-52.

9. Mrówka S. G. , W. J. Pervin, On uniform connectedness, Proc. Amer. Math. Soc., 15 (1964), 446-449.
10. Naimpally S. , Proximity Approach to Problems in Topology and Analysis, Oldenbourg Verlag, München, 2009.
11. Naimpally S., Peters J., Topology with Applications; Topological Spaces Via Near and Far, World Scientific Publishing Co. Pte. Ltd., 2013.
12. Naimpally S., Warrack B.D., Proximity Spaces, Cambridge Univ. Press, 1970.
13. Reisz F., Stetigkeitsbegriff and abstrakte Mengelehre, Atti IV Congr. Intern. dei Mat. Roma, 2 (1908), 18-24.
14. Smirnov Y.M., On Completeness of Proximity Spaces I, Amer. Math. Soc. Trans., 38 (1964), 37-73.
15. Smirnov Y.M., On Proximity Spaces, Amer. Math. Soc. Trans., 38 (1964), 5-35.
16. Szymanski P., La notion des ensembles séparés comme terme primitif de la topologie, Math. Timisoara, 17 (1941), 65-84.
17. Wallace A.D., Separation spaces, Ann. Math., 42(3) (1941), 687-697.
18. Wallace A.D., Separation spaces II, Anais. Acad. Brasil Ciencias, 14 (1942), 203-206.