Document Type : Research Paper

Authors

1 Department of Mathematics, University of Jammu, Jammu-180006, Jammu & Kashmir, India

2 Department of Mathematics and Statistics, Faculty of Sciences, P.O. 888, Taif University, Saudi Arabia

Abstract

‎For a given topological space $(X‎, ‎~\Im)$,‎ ‎there is a coarser topology on $X$ which is called the semi-regular topology on $X$ (generated by regularly open subsets) and it is denoted by $\Im^{\delta}$‎. ‎In this paper‎, ‎we study the continuity of the group operation and the inversion mapping ($\varsigma\longmapsto\varsigma^{-1}$) as regards the semi-regular topology $\Im^{\delta}$ (not necessarily with the given topology)‎. ‎Then we study the said mappings with the blend of the given topology $\Im$ and the semi-regular topology $\Im^{\delta}$‎. ‎In the twilight of this note‎, ‎we pose some questions which are noteworthy‎.

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[1] A. V. Arhangel'skii, M. Tkachenko, Topological Groups and Related Structures, Atlantis Studies in Mathematics, Vol. 1, Atlantis Press/World Scientific, Amsterdam Paris, 2008.
[2] N. Bagirmaz, I. Icen, A. Ozcan, Topological rough groups, Topol. Alegb. Appl. 4 (1) (2016), 31-38.
[3] M. Fernandez, M. Tkachenko, Subgroups of paratopological groups and feebly compact groups, Appl. Gen. Topol. 15 (2) (2014), 235-248.
[4] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly. 70 (1963), 36-41.
[5] O. Ravsky, Paratopological groups II, Matematychni Studii. 17 (2002), 93-101.
[6] S. Romaguera, M. Sanchis, M. Tkachenko, Free paratopological groups, Topology Proc. 27 (2) (2003), 613-640.
[7] M. K. Singal, S. P. Arya, On almost regular spaces, Glasnik Mat. 24 (4) (1969), 89-99.
[8] M. H. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc. 41 (1937), 375-381.
[9] N. V. Velicko, H-closed topological spaces, Amer. Math. Soc. Transl. 78 (2) (1968), 103-118.