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Year 2019, Volume: 48 Issue: 2, 490 - 500, 01.04.2019

Abstract

References

  • J. Adámek, H. Herrlich and G.E. Strecker, Abstract and Concrete Categories, John Wiley and Sons, New York, 1990.
  • M. Baran, Separation properties, Indian J. Pure Appl. Math. 23 (5), 333-341, 1992.
  • M. Baran, Separation Properties at p for the topological categories of reflexive relation spaces and preordered spaces, Math. Balkanica (N.S.) 6 (2), 193-198, 1992.
  • M. Baran, The notion of closedness in topological categories, Comment. Math. Univ. Carolin. 34 (2), 383-395, 1993.
  • M. Baran, Generalized local separation properties, Indian J. Pure Appl. Math. 25 (6), 615-620, 1994.
  • M. Baran, Separation properties in topological categories, Math. Balkanica (N.S.) 10 (1), 39-48, 1996.
  • M. Baran, A notion of compactness in topological categories, Publ. Math. Debrecen 50 (3-4), 221-234, 1997.
  • M. Baran, $T_3$ and $T_4$-objects in topological categories, Indian J. Pure Appl. Math. 29 (1), 59-69, 1998.
  • M. Baran, Completely regular objects and normal objects in topological categories, Acta Math. Hungar. 80 (3), 211-224, 1998.
  • M. Baran, Closure operators in convergence spaces, Acta Math. Hungar. 87 (1-2), 33-45, 2000.
  • M. Baran, Compactness, perfectness, separation, minimality and closedness with re- spect to closure operators, Appl. Categ. Structures 10 (4), 403-415, 2002.
  • M. Baran, Pre$T_2$-objects in topological categories, Appl. Categ. Structures 17 (6), 591-602, 2009, DOI: 10.1007/s10485-008-9161-4.
  • M. Baran and J. Al-Safar, Quotient-reflective and bireflective subcategories of the category of preordered sets, Topology Appl. 158 (15), 2076-2084, 2011, DOI 10.1016/j.topol.2011.06.043.
  • M. Baran and M. Kula, A note on separation and compactness in categories of con- vergence spaces, Proceedings of the International Conference on Applicable General Topology, Ankara, 2001, Appl. Gen. Topol. 4 (1), 1-13, 2003.
  • M. Baran and M. Kula, A note on connectedness, Publ. Math. Debrecen 68 (3-4), 489-501, 2006.
  • N. Bourbaki, General Topology, Addison-Wesley Publ. Co., 1966.
  • D. Dikranjan and E. Giuli, Closure operators I, Proceedings of the 8th international conference on categorical topology, L’Aquila, 1986, Topology Appl. 27 (2), 129-143, 1987.
  • V.A. Efremovich, Infinitesimal spaces (Russian), Doklady Akad. Nauk SSSR (N.S.) 76, 341-343, 1951.
  • L.M. Friedler, Mappings of Proximity Spaces, PH.D. Thesis, University of Alberta, 1972.
  • L.M. Friedler, Quotients of proximity spaces, Proc. Amer. Math. Soc. 37 (2), 589-594, 1973.
  • W.N. Hunsaker and P.L. Sharma, Proximity spaces and topological functors, Proc. Amer. Math. Soc. 45, 419-425, 1974.
  • P.T. Johnstone, Topos theory, London Mathematical Society Monographs, Vol. 10. Academic Press, Harcourt Brace Jovanovich, Publishers, London-New York, 1977.
  • M. Kula, A notion of connectedness in topological categories, PH.D. Thesis, University of Erciyes, 2003.
  • M. Kula, A note on Cauchy spaces, Acta Math. Hungar. 133 (1-2), 14-32, 2011, DOI: 10.1007/s10474-011-0136-9.
  • M. Kula, Separation properties at p for the topological category of Cauchy spaces, Acta Math. Hungar. 136 (1-2), 1-15, 2012, DOI: 10.1007/s10474-012-0238-z.
  • M. Kula, T. Maraşlı and S. Özkan, A note on closedness and connectedness in the category of proximity spaces, Filomat 28 (7), 1483-1492, 2014. DOI 10.2298/FIL1407483K.
  • M. Kula, S. Özkan and T. Maraşlı, Pre-Hausdorff and Hausdorff proximity spaces, Filomat 31 (12), 3837-3846, 2017.
  • S. Leader, On products of proximity spaces, Math. Ann. 154, 185-194, 1964.
  • M.W. Lodato, On topologically induced generalized proximity relations, Proc. Amer. Math. Soc. 15, 417-422, 1963.
  • M.V. Mielke, Separation axioms and geometric realizations, Indian J. Pure Appl. Math. 25 (7), 711-722, 1994.
  • J.R. Munkres, Topology: A First Course, Prentice-Hall, 1975.
  • S.A. Naimpally and B.D. Warrack, Proximity spaces, Cambridge Tracts in Mathe- matics and Mathematical Physics, No. 59 Cambridge University Press, London-New York, 1970.
  • W.J. Pervin, Quasi-proximities for topological spaces, Math. Ann. 150, 325-326, 1963.
  • G. Preuss, Theory of topological structures An approach to categorical topology, Math- ematics and its Applications, 39. D. Reidel Publishing Co., Dordrecht, 1988.
  • P.L. Sharma, Proximity bases and subbases, Pacific J. Math. 37, 515-526, 1971.
  • J.L. Sieber, Generalizations of Some Topological Notions to Syntopogenic Spaces, PhD. Thesis, The Pennslyvania State University, 1963.
  • Y.M. Smirnov, On completeness of proximity spaces I, Amer. Math. Soc. Transl. 38, 37-73, 1964.
  • Y.M. Smirnov, On proximity spaces, Mat. Sb. 31 (73), 543-574, 1952; Amer. Math. Soc. Transl. 38 (2), 5-35, 1964.
  • A.D. Wallace, Separation spaces, Ann. of Math. 42 (2), 687-697, 1941.
  • A.D. Wallace, Separation spaces II, Anais Acad. Brasil. Ci. 14, 203-206, 1942.
  • S. Willard, General Topology, Addison-Wesley, Reading, MA, 1970.

ST2, ∆T2, ST3, ∆T3, Tychonoff, compact and ∂-connected objects in the category of proximity spaces

Year 2019, Volume: 48 Issue: 2, 490 - 500, 01.04.2019

Abstract

In this paper, an explicit characterization of the separation properties $ST_2$, $\Delta T_2$, $ST_{3}$, $ \Delta T_{3} $ and Tychonoff objects are given in the topological category of proximity space. Furthermore, the (strongly) compact object and $\partial$-connected object are also characterized in the category of proximity space. Moreover, we investigate the relationships among  $ST_2$, $\Delta T_2$, $ST_{3}$, $ \Delta T_{3} $, the separation properties at a point $p$, the generalized separation properties $T_{i}$, $i=0,1,2$, $\mathbf{T_{0}}$, $\mathbf{T_{1}}$, $\mathbf{T_{2}}$ and Tychonoff objects in this category. Finally, we investigate the relationships between $\partial$-connected object and (strongly) connected object in the topological category of proximity space.

References

  • J. Adámek, H. Herrlich and G.E. Strecker, Abstract and Concrete Categories, John Wiley and Sons, New York, 1990.
  • M. Baran, Separation properties, Indian J. Pure Appl. Math. 23 (5), 333-341, 1992.
  • M. Baran, Separation Properties at p for the topological categories of reflexive relation spaces and preordered spaces, Math. Balkanica (N.S.) 6 (2), 193-198, 1992.
  • M. Baran, The notion of closedness in topological categories, Comment. Math. Univ. Carolin. 34 (2), 383-395, 1993.
  • M. Baran, Generalized local separation properties, Indian J. Pure Appl. Math. 25 (6), 615-620, 1994.
  • M. Baran, Separation properties in topological categories, Math. Balkanica (N.S.) 10 (1), 39-48, 1996.
  • M. Baran, A notion of compactness in topological categories, Publ. Math. Debrecen 50 (3-4), 221-234, 1997.
  • M. Baran, $T_3$ and $T_4$-objects in topological categories, Indian J. Pure Appl. Math. 29 (1), 59-69, 1998.
  • M. Baran, Completely regular objects and normal objects in topological categories, Acta Math. Hungar. 80 (3), 211-224, 1998.
  • M. Baran, Closure operators in convergence spaces, Acta Math. Hungar. 87 (1-2), 33-45, 2000.
  • M. Baran, Compactness, perfectness, separation, minimality and closedness with re- spect to closure operators, Appl. Categ. Structures 10 (4), 403-415, 2002.
  • M. Baran, Pre$T_2$-objects in topological categories, Appl. Categ. Structures 17 (6), 591-602, 2009, DOI: 10.1007/s10485-008-9161-4.
  • M. Baran and J. Al-Safar, Quotient-reflective and bireflective subcategories of the category of preordered sets, Topology Appl. 158 (15), 2076-2084, 2011, DOI 10.1016/j.topol.2011.06.043.
  • M. Baran and M. Kula, A note on separation and compactness in categories of con- vergence spaces, Proceedings of the International Conference on Applicable General Topology, Ankara, 2001, Appl. Gen. Topol. 4 (1), 1-13, 2003.
  • M. Baran and M. Kula, A note on connectedness, Publ. Math. Debrecen 68 (3-4), 489-501, 2006.
  • N. Bourbaki, General Topology, Addison-Wesley Publ. Co., 1966.
  • D. Dikranjan and E. Giuli, Closure operators I, Proceedings of the 8th international conference on categorical topology, L’Aquila, 1986, Topology Appl. 27 (2), 129-143, 1987.
  • V.A. Efremovich, Infinitesimal spaces (Russian), Doklady Akad. Nauk SSSR (N.S.) 76, 341-343, 1951.
  • L.M. Friedler, Mappings of Proximity Spaces, PH.D. Thesis, University of Alberta, 1972.
  • L.M. Friedler, Quotients of proximity spaces, Proc. Amer. Math. Soc. 37 (2), 589-594, 1973.
  • W.N. Hunsaker and P.L. Sharma, Proximity spaces and topological functors, Proc. Amer. Math. Soc. 45, 419-425, 1974.
  • P.T. Johnstone, Topos theory, London Mathematical Society Monographs, Vol. 10. Academic Press, Harcourt Brace Jovanovich, Publishers, London-New York, 1977.
  • M. Kula, A notion of connectedness in topological categories, PH.D. Thesis, University of Erciyes, 2003.
  • M. Kula, A note on Cauchy spaces, Acta Math. Hungar. 133 (1-2), 14-32, 2011, DOI: 10.1007/s10474-011-0136-9.
  • M. Kula, Separation properties at p for the topological category of Cauchy spaces, Acta Math. Hungar. 136 (1-2), 1-15, 2012, DOI: 10.1007/s10474-012-0238-z.
  • M. Kula, T. Maraşlı and S. Özkan, A note on closedness and connectedness in the category of proximity spaces, Filomat 28 (7), 1483-1492, 2014. DOI 10.2298/FIL1407483K.
  • M. Kula, S. Özkan and T. Maraşlı, Pre-Hausdorff and Hausdorff proximity spaces, Filomat 31 (12), 3837-3846, 2017.
  • S. Leader, On products of proximity spaces, Math. Ann. 154, 185-194, 1964.
  • M.W. Lodato, On topologically induced generalized proximity relations, Proc. Amer. Math. Soc. 15, 417-422, 1963.
  • M.V. Mielke, Separation axioms and geometric realizations, Indian J. Pure Appl. Math. 25 (7), 711-722, 1994.
  • J.R. Munkres, Topology: A First Course, Prentice-Hall, 1975.
  • S.A. Naimpally and B.D. Warrack, Proximity spaces, Cambridge Tracts in Mathe- matics and Mathematical Physics, No. 59 Cambridge University Press, London-New York, 1970.
  • W.J. Pervin, Quasi-proximities for topological spaces, Math. Ann. 150, 325-326, 1963.
  • G. Preuss, Theory of topological structures An approach to categorical topology, Math- ematics and its Applications, 39. D. Reidel Publishing Co., Dordrecht, 1988.
  • P.L. Sharma, Proximity bases and subbases, Pacific J. Math. 37, 515-526, 1971.
  • J.L. Sieber, Generalizations of Some Topological Notions to Syntopogenic Spaces, PhD. Thesis, The Pennslyvania State University, 1963.
  • Y.M. Smirnov, On completeness of proximity spaces I, Amer. Math. Soc. Transl. 38, 37-73, 1964.
  • Y.M. Smirnov, On proximity spaces, Mat. Sb. 31 (73), 543-574, 1952; Amer. Math. Soc. Transl. 38 (2), 5-35, 1964.
  • A.D. Wallace, Separation spaces, Ann. of Math. 42 (2), 687-697, 1941.
  • A.D. Wallace, Separation spaces II, Anais Acad. Brasil. Ci. 14, 203-206, 1942.
  • S. Willard, General Topology, Addison-Wesley, Reading, MA, 1970.
There are 41 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Muammer Kula 0000-0002-1366-6149

Publication Date April 1, 2019
Published in Issue Year 2019 Volume: 48 Issue: 2

Cite

APA Kula, M. (2019). ST2, ∆T2, ST3, ∆T3, Tychonoff, compact and ∂-connected objects in the category of proximity spaces. Hacettepe Journal of Mathematics and Statistics, 48(2), 490-500.
AMA Kula M. ST2, ∆T2, ST3, ∆T3, Tychonoff, compact and ∂-connected objects in the category of proximity spaces. Hacettepe Journal of Mathematics and Statistics. April 2019;48(2):490-500.
Chicago Kula, Muammer. “ST2, ∆T2, ST3, ∆T3, Tychonoff, Compact and ∂-Connected Objects in the Category of Proximity Spaces”. Hacettepe Journal of Mathematics and Statistics 48, no. 2 (April 2019): 490-500.
EndNote Kula M (April 1, 2019) ST2, ∆T2, ST3, ∆T3, Tychonoff, compact and ∂-connected objects in the category of proximity spaces. Hacettepe Journal of Mathematics and Statistics 48 2 490–500.
IEEE M. Kula, “ST2, ∆T2, ST3, ∆T3, Tychonoff, compact and ∂-connected objects in the category of proximity spaces”, Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 2, pp. 490–500, 2019.
ISNAD Kula, Muammer. “ST2, ∆T2, ST3, ∆T3, Tychonoff, Compact and ∂-Connected Objects in the Category of Proximity Spaces”. Hacettepe Journal of Mathematics and Statistics 48/2 (April 2019), 490-500.
JAMA Kula M. ST2, ∆T2, ST3, ∆T3, Tychonoff, compact and ∂-connected objects in the category of proximity spaces. Hacettepe Journal of Mathematics and Statistics. 2019;48:490–500.
MLA Kula, Muammer. “ST2, ∆T2, ST3, ∆T3, Tychonoff, Compact and ∂-Connected Objects in the Category of Proximity Spaces”. Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 2, 2019, pp. 490-0.
Vancouver Kula M. ST2, ∆T2, ST3, ∆T3, Tychonoff, compact and ∂-connected objects in the category of proximity spaces. Hacettepe Journal of Mathematics and Statistics. 2019;48(2):490-50.