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. 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.

Anahtar Kelimeler:

Zaman Skalası, de Groot Dual Topolojisi, Topolojik Özellikler

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 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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[1] J. de Groot, G.E. Strecker and E. Wattel, The compactness operator in general topology, in: Proceedings of the Second Prague Topological Symposium, Prague, 1966, 161–163.
[2] J. de Groot, An isomorphism principle in general topology, Bull. Amer. Math. Soc., 73 (1967) 465–467.
[3] R. Kopperman, Asymmetry and duality in topology, Topology Appl. 66 (1) (1995) 1–39.
[4] M. M. Kovár, At most 4 topologies can arise from iterating the de Groot dual, Topology Appl. 130 (1) (2003) 175–182.
[5] M. M. Kovár, On iterated de Groot dualizations of topological spaces, Topology Appl. 146 (147) (2005), 83–89.
[6] M. M. Kovár, A new causal topology and why the universe is co-compact, arXiv:1112.0817 [math-ph], 2013.
[7] J. D. Lawson, M. Mislove, Problems in domain theory and topology, in: J. Van Mill, G.M. Reed (Eds.), Open Problems in Topology, North-Holland, Amsterdam, 1990, pp. 349–372.
[8] N. Liden, Spaces, Their Anti-spaces and Related Maps., Washington Uni., Phd Thesis, (1973).
[9] T. Yokoyama, A counterexample for some problem for de Groot dual iterations. Topology Appl. 156 (2009), no. 13, 2224–2225.
[10] S. Hilger, Ein Maßkettenkalkul mit Anwendung auf Zentrumsmannigfaltigkeiten. Ph.D. Thesis, Universität Würzburg, Würzburg, 1988.
[11] B. Aulbach and S. Hilger, A uniﬁed approach to continuous and discrete dynamics, Qualitative Theory of Differential Equations (Szeged, 1988), Colloq. Math. Soc. Janos. Bolyai. North Holland, Amsterdam, (1990)37–56.
[12] S. Hilger, Analysis on measure chains—a unified approach to continuous and discrete calculus, Results Math. 18 (1990), no. 1-2, 18–56.
[13] M. Bohner and A. Peterson, Dynamic equations on time scale. An introduction with applications. Birkhauser, Boston, 2001.
[14] M. Bohner and A. Peterson, Advances in dynamic equations on time scales. Birkhauser, Boston, 2003.
[15] R. P. Agarwal, M. Bohner, Basic calculus on time scale and some of its applications, Results Math. 35 (1999), no. 1-2, 3–22.
[16] G. Sh. Guseinov, Integration on time scale, J. Math. Anal. Appl. 285 (2003), 107-127.
[17] T. V. Gray, Opial’s inequality on time scales and an application, Georgia Southern University, Electronic Theses & Dissertations, (2007), Paper 652.
[18] N. Esty and S. Hilger, Convergence of time scales under the fell topology, J. Difference Equ. Appl. 15 (2009), no. 10, 1011–1020.
[19] R. Oberste-Vorth, The Fell topology on the space of time scale for dynamic equations, Adv. Dyn. Syst. Appl. 3 (2008), no. 1, 177–184.