Some of the Topological Gaps of a Generalized Metric Space

Subject Areas : Statistics

Yousef  Sohooly ¹ (Depth. of Math.,Islamic Azad Univ. -Shiraz Branch, Shiraz-Iran.)
Khadijeh  Jahedi ² (Dept. of Math. Islamic Azad University-Shiraz Branch, Shiraz-Iran)

Received: 2020-12-28 Accepted : 2021-08-27 Published : 2022-10-23

Keywords: همگرایی آماری و چگالی طبیعی, توپولوژی تعمیم‌یافته به سبک سازا, واژه‌های کلیدی: متریک, توپولوژی, متریک تعمیم‌یافته,

Abstract :

‎In this paper‎, ‎we introduce the neighborhood of each element x ‎of the Jleli_Samet's generalized metric space (X,d) ‎and ‎we ‎study ‎some ‎properties ‎of ‎the collection T_d of all unions of such neighborhoods which is called generalized topology in the sence of Csaszar. We show that some of the elementary properties on the metric spaces dose not happen on the generalized metric space in the sense of Jleli_Samet. Moreover the generalized topology T_d need not to be compatible. Also we give a sufficient condition to show that any point of the generalized metric space in the sense of Jleli_‎Samet‎, ‎has zero self-distance‎. ‎We introduce the statistical generalized metric space by replacing the concept of convergence with statistical convergence on the definition of generalized metric space in the sense of Jleli_ Samet and we show that two generalized metric spaces are equivalent.||| ||| ||| ||| ||| ||| ||| ‎

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