Zaman skalalarının de Groot Duali

Year 2017, Volume: 21 Issue: 6, 1336 - 1341, 01.12.2017

Soley Ersoy , Hilal Polat Ayşenur Türkoğlu

https://doi.org/10.16984/saufenbilder.297047

Abstract

Bu çalışmada, zaman skalasının de Groot dual
topolojisini inceledik. De Groot dual topolojisi fiili sonsuzluk yerine
potansiyel sonsuzluk ile ilgilidir.  reel sayı doğrusu
zamanı göstermek üzere onun de Groot duali olan  kompakttır ve zamanın sınırsız ancak
kompaktlık açısında sonlu olduğu fikrini verir. Diğer taraftan zaman skalaları
da sadece reel aralıklar veya ayrık kümeleri değil ’nin tüm kapalı alt kümeleridir ve reel
sayıları da içermektedir.  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.

Keywords

Zaman Skalası, de Groot Dual Topolojisi, Topolojik Özellikler

The de Groot Dual of time scales

Abstract

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

Keywords

Time-scale, de Groot Dual Topology, Topological Properties

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