Soley ERSOY, Hilal POLAT, Ayşenur TÜRKOĞLU

Zaman skalalarının de Groot Duali

Bu çalışmada, zaman skalasının de Groot dual topolojisini inceledik. De Groot dual topolojisi fiili sonsuzluk yerine potansiyel sonsuzluk ile ilgilidir. ℝ reel sayı doğrusu zamanı göstermek üzere onun de Groot duali olan * ℝ kompakttır ve zamanın sınırsız ancak kompaktlık açısında sonlu olduğu fikrini verir. Diğer taraftan zaman skalaları da sadece reel aralıklar veya ayrık kümeleri değil ℝ ’nin tüm kapalı alt kümeleridir ve reel sayıları da içermektedir. * ℝ tüm sınırlı zaman skaları üzerinde alışılmış topolojiye sahip olur fakat zaman skalaları sınırsız iken topolojik yapısı farklılaşır. Bu nedenle zaman skalasının de Groot dual topolojisine göre topolojik özelliklerini inceledik ve bağlantılılık koşullarını belirledik. Ayrıca sonuçlarımızı bilinen ayrık ve sürekli zaman skalaları ile örneklendirdik.

The de Groot Dual of time scales

In this paper, we investigate the de Groot dual topology of time scales. The de Groot dual topology is relatedto the concept of potential infinity instead of actual infinity. Whenever the real number line ℝ denotes timethen its dual space * ℝ is compact and this provides insight that time is unbounded but finite in the sense ofcompact. On the other hand time scales are arbitrary non-empty closed subsets of ℝ (not only the realintervals or discrete sets) and include the real numbers. * ℝ has the usual topology on every bounded timescales but its topological structure differs when time scales are unbounded. Therefore, we state thetopological properties of a time scale with respect the de Groot dual topology and determine theconnectedness conditions of it. Moreover, we illustrate our results with known examples of discrete andcontinuous time scales.

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