Abstract
We investigate weak and strong structures for generalized topological spaces, among others products, sums, subspaces, quotients, and the complete lattice of generalized topologies on a given set. Also we introduce \({T_{3.5}}\) generalized topological spaces and give a necessary and sufficient condition for a generalized topological space to be a \({T_{3.5}}\) space: they are exactly the subspaces of powers of a certain natural generalized topology on [0,1]. For spaces with at least two points here we can have even dense subspaces. Also, \({T_{3.5}}\) generalized topological spaces are exactly the dense subspaces of compact \({T_4}\) generalized topological spaces. We show that normality is productive for generalized topological spaces. For compact generalized topological spaces we prove the analogue of the Tychonoff product theorem. We prove that also Lindelöfness (and \({\kappa}\)-compactness) is productive for generalized topological spaces. On any ordered set we introduce a generalized topology and determine the continuous maps between two such generalized topological spaces: for \({|X|, |Y| \geqq 2}\) they are the monotonous maps continuous between the respective order topologies. We investigate the relation of sums and subspaces of generalized topological spaces to ways of defining generalized topological spaces.
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Abd El-Monsef M. E., El-Deeb S. N., Mahmoud R. A.: \({\beta}\)-open sets and \({\beta}\)-continuous mappings. Bull. Fac. Sci. Assiut Univ. A 12, 77–90 (1983)
J. Adámek, H. Herrlich and G. E. Strecker, Abstract and Concrete Categories: the Joy of Cats, Reprint of the 1990 original, Wiley (New York), Repr. Theory Appl. Categ. 17 (2006).
Arar M.: On countably \({\mu}\)-paracompact spaces. Acta Math. Hungar. 146, 50–57 (2016)
T. Bonnesen and W. Fenchel, Theorie der konvexen Körper, Berichtigter Reprint, Springer (Berlin etc., 1974); English transl.: Theory of Convex Bodies, Transl. from German and ed. by L. Boron, C. Christenson, B. Smith, with collab. of W. Fenchel, BCS Associates (Moscow, ID, 1987).
G. Castellini, Categorical Closure Operators, Math.: Theory and Appl., Birkhäuser (Boston, MA, 2003).
E. Čech, Topological Spaces, revised edition by Z. Frolík and M. Katětov, Scientific ed. V. Pták, Ed. of the English transl. C. O. Junge, Publishing House of Czechoslovak Acad. Sci. (Praha), Interscience, Wiley (London etc., 1966).
Császár Á.: Generalized open sets. Acta Math. Hungar. 75, 65–87 (1997)
Császár Á.: Generalized topology, generalized continuity. Acta Math. Hungar. 96, 351–357 (2002)
Császár Á.: Separation axioms for generalized topologies. Acta Math. Hungar. 104, 63–69 (2004)
Császár Á.: Generalized open sets in generalized topologies. Acta Math. Hungar. 106, 53–66 (2005)
Császár Á.: \({\lambda}\)-open sets in generalized topological spaces. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 49, 59–63 (2006)
Császár Á.: Normal generalized topologies. Acta Math. Hungar. 115, 309–313 (2007)
Császár Á.: Enlargements and generalized topologies. Acta Math. Hungar. 120, 351–354 (2008)
Császár Á.: On generalized neighbourhood systems. Acta Math. Hungar. 121, 395–400 (2008)
Császár Á.: Product of generalized topologies. Acta Math. Hungar. 123, 127–132 (2009)
Császár Á., Makai E. Jr.: Further remarks on \({\delta}\)- and \({\theta}\)-modifications. Acta Math. Hungar. 123, 223–228 (2009)
D. Dikranjan, E. Giuli, Closure operators. I, in: Proc. 8th Internat. Conf. on Categ. Top. (L’Aquila, 1986), Topology Appl., 27 (1987), pp. 129–143.
D. Dikranjan, E. Giuli and W. Tholen, Closure operators. II, in: Categorical Topology and its Relations to Analysis, Algebra and Combinatorics (Prague, 1988), World Sci. Publ. (Teaneck, NJ, 1989), pp. 287–335.
D. Dikranjan and W. Tholen, Categorical Structure of Closure Operators, With Applications to Topology, Algebra and Discrete Mahematics, Math. and its Appl. 346, Kluwer (Dordrecht, 1995).
N. Dunford and J. T. Schwartz, Linear Operators. I. General Theory, with the assistance of W. G. Bade and R. G. Bartle, Pure and Appl. Math. 7, Interscience Inc., (New York–London, 1958).
Encyclopaedia of Mathematics, vol. 9, Translated from Russian, updated, Ed. M. Hazewinkel, Kluwer (Dordrecht, etc., 1993), Topological structures, pp. 197–204.
R. Engelking, General Topology, Volume 6 of Sigma Series in Pure Mathematics, Heldermann Verlag (Berlin, 1989).
Ge X., Ge Y.: \({\mu}\)-separations in generalized topological spaces. Appl. Math. J. Chinese Univ. Ser. B 25, 243–252 (2010)
H. Herrlich and G. E. Strecker, Category Theory. An introduction, third ed., Sigma Series in Pure Math. 1, Heldermann (Lemgo, 2007).
I. Juhász, Cardinal Functions in Topology — Ten Years Later, 2nd ed., Math. Centre Tracts 123, Math. Centrum (Amsterdam, 1980).
Kent D. C., Min W. K.: Neighborhood spaces. Int. J. Math. Math. Sci. 32, 387–399 (2002)
Kim Y. K., Min W. K.: Remarks on enlargements of generalized topologies. Acta Math. Hungar. 130, 390–395 (2011)
Levine N.: Semi-open sets and semi-continuity in topological spaces Amer. Math. Monthly 70, 36–41 (1963)
Lugojan S.: Generalized topology (Romanian. English summary) Stud. Cerc. Mat. 34, 348–360 (1982)
Mashhour A. S., El-Monsef Abd, El-Deeb S. N.: On precontinuous and weak precontinuous mappings. Proc. Math. Phys. Soc. Egypt 53, 47–53 (1982)
Mashhour A. S., Allam A. A., Hasanein I. A.: Bisupratopological spaces. Proc. Math. Phys. Soc. Egypt 62, 39–47 (1986)
Mashhour A. S., Allam A. A., Mahmoud F. S., Khedr F. H.: On supratopological spaces. Indian J. Pure Appl. Math. 14, 502–510 (1983)
Min W.K.: Remarks on separation axioms on generalized topological spaces. J. Chungcheong Math. Soc. 23, 293–298 (2010)
Min W.K.: On ascending generalized neighbourhood spaces. Acta Math. Hungar. 127, 391–396 (2010)
Njåstad O.: On some classes of nearly open sets. Pacific J. Math. 15, 961–970 (1965)
Noble N.: Products with closed projections. II. Trans. Amer. Math. Soc. 160, 169–183 (1971)
Pavlović V., Cvetković A.S.: On generalized topologies arising from mappings. Bull. Iranian Math. Soc. 38, 553–565 (2012)
G. Preuss, Theory of Topological Structures, Math. and its Appl. 39, Reidel (Dordrecht, 1988).
Sarsak M.S.: Weakly \({\mu}\)-compact spaces. Demonstratio Math. 45, 929–938 (2012)
Sarsak M.S.: On \({\mu}\)-compact sets in \({\mu}\)-spaces, Questions Answers Gen.. Topology 31, 49–57 (2013)
R. Schneider, Convex bodies: the Brunn–Minkowski Theory, Second expanded edition, Encyclopedia of Math. and its Appl. 44, 151, Cambridge Univ. Press (Cambridge, 1993, 2014).
Shen R.: Remarks on products of generalized topologies. Acta Math. Hungar. 124, 363–369 (2009)
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Makai, E., Peyghan, E. & Samadi, B. Weak and strong structures and the \({T_{3.5}}\) property for generalized topological spaces. Acta Math. Hungar. 150, 1–35 (2016). https://doi.org/10.1007/s10474-016-0653-7
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DOI: https://doi.org/10.1007/s10474-016-0653-7
Key words and phrases
- generalized topology
- weak and strong structure
- product
- sum
- subspace
- quotient
- \({T_{3.5}}\)
- normal
- compact
- Lindelöf
- \({\kappa }\)-compact
- ordered generalized topological space