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New Conceptions of Transitivity and Minimal Mappings

Received: 13 December 2013    Accepted:     Published: 30 January 2014
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Abstract

The concepts of topological δ- transitive maps, α-type transitive maps, δ-minimal and α-minimal mappings were introduced by M. Nokhas Murad Kaki. In this paper, the relationship between two different notions of transitive maps, namely topological δ-type transitive mapsandtopological α-type transitive maps has been studied and some of their properties in two topological spaces (X, τδ)and (X, τα), τδ denotes the δ–topology (resp. τα denotes the α–topology) of a given topological space (X, τ) has been investigated.. Also, we have proved that there exists a dense orbit in X, where X is locally compact Hausdorff space and τ has a countable basis. The main results are the following propositions:Every topologically α-type transitive map is a topologically transitive map which implies topologically δ- transitive map, but the converse not necessarily true., and every α-minimal map is a minimal map which implies δ- minimal map in topological spaces, but the converse not necessarily true. Finally, we have proved that a map which is γr- conjugated to γ-transitive (resp. γ-minimal, γ-mixing) map is γ-transitive (resp. γ-minimal, γ-mixing).

DOI 10.11648/j.sr.20140201.11
Published in Science Research (Volume 2, Issue 1, February 2014)
Page(s) 1-6
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Topologically δ-Transitive, δ-Irresolute, δ-Type Transitive,δ-Dense, γ-Dense, γ Transitive

References
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[3] N. V. Velicko, H-closed topological spaces. Amer. Math. Soc. Transl. 1968, Vol. 78, p102-118.
[4] Mohammed Nokhas Murad, New Types of δ-Transitive Maps, International Journal of Engineering & Technology IJET-IJENS Vol:12 No:06, pp.134-136.
[5] Levine N., Semi open sets and semi continuity in topological spaces. Amer. Math. Monthly.1963, Vol.70, p 36- 41.
[6] Bhattacharya P., and Lahiri K.B., Semi-generalized closed sets in topology. Indian J. Math. , 1987, Vol. 29, p376-382.
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[8] Kasahara S., Operation-compact spaces. MathematicaJaponica, 1979, Vol. 24, p97-105.
[9] M. Caldas, S. Jafari and M. M. Kovar, Some properties ofθ-open sets, Divulge. Mat, 12(2)(2004), p 161-169.
[10] Caldas M., A note on some applications of α-open sets, UMMS, 2003, Vol. 2, p125-130.
[11] Mohammed Nokhas Murad, Topologically - Transitive Maps and Minimal Systems Gen. Math. Notes, 2012, Vol. 10, No. 2, pp. 43-53 ISSN 2219-7184; Copyright © ICSRS
[12] Maheshwari N. S., and Thakur S. S., On α-irresolute mappings, Tamkang J. Math, 1980, Vol. 11, p209-214.
[13] Ogata N., On some classes of nearly open sets, Pacific J. Math, 1965, Vol. 15, p 961-970..
[14] F.H. Khedr and T. Noiri.On θ-irresolute functions. Indian J. Math., 1986, Vol. 3, No:28, p 211-217.
[15] M. Nokhas Murad Kaki, Introduction to θ -Type Transitive Maps on Topological spaces.International Journal of Basic & Applied Sciences IJBAS-IJENS 2012, Vol:12, No:06 p 104-108
[16] Andrijevie D., Some properties of the topology of α-sets,Math. Vesnik, 1994, p 1 -10
[17] Arenas G. F., Dontchev J. andPuertas L.M.Some covering properties of the α-topology , 1998.
[18] Caldas M. and Dontchev J., On space with hereditarily compact α-topologies, Acta. Math. Hung, 1999. Vol. 82, p121-129.
[19] M. Nokhas Murad Kaki,ON SOME NEW γ -TYPE MAPS ON TOPOLOGICAL SPACES,Journal of Mathematical Sciences: Advances and Applications), 2013, Vol. 20 p. 45-60
[20] M. Nokhas Murad Kaki, Relationship between New Types of Transitive Maps and Minimal Systems International Journal of Electronics Communication and Computer Engineering, 2013, Volume 4, Issue 6, p. 2278–4209
Author Information
  • Mathematics Department, School of Science, Faculty of Science & Science Education, University of Sulaimani, Kurdistan Region-Iraq

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    Mohammed Nokhas Murad Kaki. (2014). New Conceptions of Transitivity and Minimal Mappings. Science Research, 2(1), 1-6. https://doi.org/10.11648/j.sr.20140201.11

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    Mohammed Nokhas Murad Kaki. New Conceptions of Transitivity and Minimal Mappings. Sci. Res. 2014, 2(1), 1-6. doi: 10.11648/j.sr.20140201.11

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    Mohammed Nokhas Murad Kaki. New Conceptions of Transitivity and Minimal Mappings. Sci Res. 2014;2(1):1-6. doi: 10.11648/j.sr.20140201.11

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  • @article{10.11648/j.sr.20140201.11,
      author = {Mohammed Nokhas Murad Kaki},
      title = {New Conceptions of Transitivity and Minimal Mappings},
      journal = {Science Research},
      volume = {2},
      number = {1},
      pages = {1-6},
      doi = {10.11648/j.sr.20140201.11},
      url = {https://doi.org/10.11648/j.sr.20140201.11},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.sr.20140201.11},
      abstract = {The concepts of topological δ- transitive maps, α-type transitive maps, δ-minimal and α-minimal mappings were introduced by M. Nokhas Murad Kaki. In this paper, the relationship between two different notions of transitive maps, namely topological δ-type transitive mapsandtopological α-type transitive maps has been studied and some of their properties in two topological spaces (X, τδ)and (X, τα), τδ denotes the δ–topology (resp. τα denotes the α–topology) of a given topological space (X, τ) has been investigated..  Also, we have proved that there exists a dense orbit in X, where X is locally compact Hausdorff space and τ has a countable basis. The main results are the following propositions:Every topologically α-type transitive map is a topologically transitive map which implies topologically δ- transitive map, but the converse not necessarily true., and every α-minimal map is a minimal map which implies δ- minimal map in topological spaces, but the converse not necessarily true. Finally, we have proved that a map which is γr- conjugated to γ-transitive (resp. γ-minimal, γ-mixing) map is γ-transitive (resp. γ-minimal, γ-mixing).},
     year = {2014}
    }
    

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  • TY  - JOUR
    T1  - New Conceptions of Transitivity and Minimal Mappings
    AU  - Mohammed Nokhas Murad Kaki
    Y1  - 2014/01/30
    PY  - 2014
    N1  - https://doi.org/10.11648/j.sr.20140201.11
    DO  - 10.11648/j.sr.20140201.11
    T2  - Science Research
    JF  - Science Research
    JO  - Science Research
    SP  - 1
    EP  - 6
    PB  - Science Publishing Group
    SN  - 2329-0927
    UR  - https://doi.org/10.11648/j.sr.20140201.11
    AB  - The concepts of topological δ- transitive maps, α-type transitive maps, δ-minimal and α-minimal mappings were introduced by M. Nokhas Murad Kaki. In this paper, the relationship between two different notions of transitive maps, namely topological δ-type transitive mapsandtopological α-type transitive maps has been studied and some of their properties in two topological spaces (X, τδ)and (X, τα), τδ denotes the δ–topology (resp. τα denotes the α–topology) of a given topological space (X, τ) has been investigated..  Also, we have proved that there exists a dense orbit in X, where X is locally compact Hausdorff space and τ has a countable basis. The main results are the following propositions:Every topologically α-type transitive map is a topologically transitive map which implies topologically δ- transitive map, but the converse not necessarily true., and every α-minimal map is a minimal map which implies δ- minimal map in topological spaces, but the converse not necessarily true. Finally, we have proved that a map which is γr- conjugated to γ-transitive (resp. γ-minimal, γ-mixing) map is γ-transitive (resp. γ-minimal, γ-mixing).
    VL  - 2
    IS  - 1
    ER  - 

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