saujs Sakarya University Journal of Science 2147-835X Sakarya University 10.16984/saufenbilder.297047 Mathematical Sciences Matematik Zaman skalalarının de Groot Duali The de Groot Dual of time scales https://orcid.org/0000-0002-7183-7081 Ersoy Soley Sakarya University Polat Hilal Türkoğlu Ayşenur 12 01 2017 21 6 1336 1341 03 09 2017 08 01 2017 Copyright © 1997, Sakarya University Journal of Science 1997 Sakarya University Journal of Science

Bu çalışmada, zaman skalasının de Groot dualtopolojisini inceledik. De Groot dual topolojisi fiili sonsuzluk yerinepotansiyel sonsuzluk ile ilgilidir. reel sayı doğrusuzamanı göstermek üzere onun de Groot duali olan kompakttır ve zamanın sınırsız ancakkompaktlık açısında sonlu olduğu fikrini verir. Diğer taraftan zaman skalalarıda sadece reel aralıklar veya ayrık kümeleri değil ’nin tüm kapalı alt kümeleridir ve reelsayıları da içermektedir. tüm sınırlı zaman skaları üzerinde alışılmıştopolojiye sahip olur fakat zaman skalaları sınırsız iken topolojik yapısıfarklılaşır. Bu nedenle zaman skalasının de Groot dual topolojisine göretopolojik özelliklerini inceledik ve bağlantılılık koşullarını belirledik. Ayrıca sonuçlarımızı bilinen ayrık vesürekli zaman skalaları ile örneklendirdik.

In this paper, we investigate the de Groot dualtopology of time scales. The de Groot dual topology is related to the conceptof potential infinity instead of actual infinity. Whenever the real number linedenotes time thenits dual space is compact andthis provides insight that time is unbounded but finite in the sense ofcompact. On the other hand time scales are arbitrary non-empty closed subsetsof (not only the realintervals or discrete sets) and include the real numbers. has the usualtopology on every bounded time scales but its topological structure differswhen time scales are unbounded. Therefore, we state the topological propertiesof a time scale with respect the de Groot dual topology and determine theconnectedness conditions of it. Moreover, we illustrate our results with knownexamples of discrete and continuous time scales.

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