Abstract
In this paper, we establish several useful consequences of the following, and some other closely related, basic definitions introduced in some former papers by the first author. A family \( \mathcal{R} \) of relations on one set X to another Y is called a relator on X to Y. Moreover, the ordered pair \( (\,X\,,\,Y \,)(\,\mathcal{R}\,) ={\bigl (\, (\,X\,,\,Y \,),\ \mathcal{R}\,\bigr )} \) is called a relator space. A function \( \square \) of the class of all relator spaces to the class of all relators is called a direct unary operation for relators if, for any relator \( \mathcal{R} \) on X to Y, the value \( \mathcal{R}^{\,\,\square } = \mathcal{R}^{\ \square _{X\,Y }} = \square \,{\bigl ((\,X,\,Y \,)(\,\mathcal{R}\,)\bigr )} \) is also relator on X to Y. If \( (\,X\,,\,Y \,)(\,\mathcal{R}\,) \) and \( (\,Z\,,\,W\,)(\,\mathcal{S}\,) \) are relator spaces and \( \square \) is a direct unary operation for relators, then a pair \( (\,\mathcal{F}\,,\ \mathcal{G}\,) \) of relators \( \mathcal{F} \) on X to Z and \( \mathcal{G} \) on Y to W is called mildly \( \square \)–continuous if, under the elementwise inversion and compositions of relators, we have \( \bigl ((\mathcal{G}^{\,\square }\,)^{-1}\! \circ \,\mathcal{S}^{\,\square }\circ \,\mathcal{F}^{\,\,\square }\,\bigr )^{\square }\subseteq \mathcal{R}^{\ \square \,\square } \).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Al-Omari, A.: Contra continuity on weak structure spaces. Rend. Inst. Mat. Univ. Trieste 44, 423–437 (2012)
Bourbaki, N.: General Topology, Chaps. 1–4. Springer, Berlin (1989)
Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (2002)
Davis, A.S.: Indexed systems of neighborhoods for general topological spaces. Am. Math. Monthly 68, 886–893 (1961)
Dontchev, J.: Contra-continuous functions and strongly S-closed spaces. Int. J. Math. Sci. 19, 303–310 (1996)
Efremovič, V.A.: The geometry of proximity. Mat. Sb. 31, 189–200 (1952) (Russian)
Efremović, V.A., Švarc, A.S.: A new definition of uniform spaces. Metrization of proximity spaces. Dokl. Acad. Nauk. SSSR 89, 393–396 (1953) (Russian)
Ekici, E., Jafari, S., Noiri, T.: On upper and lower contra-continuous multifunctions. An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 54, 75–85 (2008)
Fletcher, P., Lindgren, W.F.: Quasi-Uniform Spaces. Marcel Dekker, New York (1982)
Ganter, B., Wille, R.: Formal Concept Analysis. Springer, Berlin (1999)
Glavosits, T.: Generated preorders and equivalences. Acta Acad. Paed. Agriensis Sect. Math. 29, 95–103 (2002)
Jayanthi, D.: Contra continuity on generalized topological spaces. Acta Math. Hungar. 137, 263–271 (2012)
Kelley, J.L.: General Topology. Van Nostrand Reinhold Company, New York (1955)
Kenyon, H.: Two theorems about relations. Trans. Am. Math. Soc. 107, 1–9 (1963)
Kurdics, J., Száz, Á.: Well-chained relator spaces. Kyungpook Math. J. 32, 263–271 (1992)
Levine, N.: On uniformities generated by equivalence relations. Rend. Circ. Mat. Palermo 18, 62–70 (1969)
Levine, N.: On Pervin’s quasi uniformity. Math. J. Okayama Univ. 14, 97–102 (1970)
Li, Z., Zhu, W.: Contra continuity on generalized topological spaces. Acta Math. Hungar. 138, 34–43 (2013)
Mala, J.: Relators generating the same generalized topology. Acta Math. Hungar. 60, 291–297 (1992)
Mala, J., Száz, Á.: Properly topologically conjugated relators. Pure Math. Appl. Ser. B 3, 119–136 (1992)
Mala, J., Száz, Á.: Modifications of relators. Acta Math. Hungar. 77, 69–81 (1997)
Nakano, H., Nakano, K.: Connector theory. Pac. J. Math. 56, 195–213 (1975)
Noiri, T., Popa, V.: A unified theory of contra-continuity for functions. Ann. Univ. Sci. Budapest 44, 115–137 (2002)
Pataki, G.: Supplementary notes to the theory of simple relators. Radovi Mat. 9, 101–118 (1999)
Pataki, G.: On the extensions, refinements and modifications of relators. Math. Balk. 15, 155–186 (2001)
Pataki, G., Száz, Á.: A unified treatment of well-chainedness and connectedness properties. Acta Math. Acad. Paedagog. Nyházi. (N.S.) 19, 101–165 (2003)
Pervin, W.J.: Quasi-uniformization of topological spaces. Math. Ann. 147, 316–317 (1962)
Rakaczki, Cs., Száz, Á.: Semicontinuity and closedness properties of relations in relator spaces. Mathematica (Cluj) 45, 73–92 (2003)
Smirnov, Yu.M.: On proximity spaces. Math. Sb. 31, 543–574 (1952) (Russian)
Stromberg, K.R.: An Introduction to Classical Real Analysis. Wadsworth, Belmont (1981)
Száz, Á.: Basic tools and mild continuities in relator spaces. Acta Math. Hungar. 50, 177–201 (1987)
Száz, Á.: Directed, topological and transitive relators. Publ. Math. Debrecen 35, 179–196 (1988)
Száz, Á.: Projective and inductive generations of relator spaces. Acta Math. Hungar. 53, 407–430 (1989)
Száz, Á.: Relators, nets and integrals. Unfinished doctoral thesis, Debrecen, 126 pp. (1991)
Száz, Á.: The fat and dense sets are more important than the open and closed ones. In: Abstracts of the Seventh Prague Topological Symposium, Inst. Math. Czechoslovak Acad. Sci., p. 106 (1991)
Száz, Á.: Inverse and symmetric relators. Acta Math. Hungar. 60, 157–176 (1992)
Száz, Á.: Structures derivable from relators. Singularité 3, 14–30 (1992)
Száz, Á.: Refinements of relators. Tech. Rep., Inst. Math., Univ. Debrecen 76, 19 pp. (1993)
Száz, Á.: Topological characterizations of relational properties. Grazer Math. Ber. 327, 37–52 (1996)
Száz, Á.: Uniformly, proximally and topologically compact relators. Math. Pannon. 8, 103–116 (1997)
Száz, Á.: An extension of Kelley’s closed relation theorem to relator spaces. Filomat 14, 49–71 (2000)
Száz, Á.: Somewhat continuity in a unified framework for continuities of relations. Tatra Mt. Math. Publ. 24, 41–56 (2002)
Száz, Á.: Upper and lower bounds in relator spaces. Serdica Math. J. 29, 239–270 (2003)
Száz, Á.: Lower and bounds in ordered sets without axioms. Tech. Rep., Inst. Math., Univ. Debrecen 2004/1, 11 pp. (2004)
Száz, Á.: Rare and meager sets in relator spaces. Tatra Mt. Math. Publ. 28, 75–95 (2004)
Száz, Á.: Galois-type connections on power sets and their applications to relators. Tech. Rep., Inst. Math., Univ. Debrecen 2005/2, 38 pp. (2005)
Száz, Á.: Minimal structures, generalized topologies, and ascending systems should not be studied without generalized uniformities. Filomat 21, 87–97 (2007)
Száz, Á.: Galois type connections and closure operations on preordered sets. Acta Math. Univ. Comen. 78, 1–21 (2009)
Száz, Á.: Foundations of the theory of vector relators. Adv. Stud. Contemp. Math. (Kyungshang) 20, 139–195 (2010)
Száz, Á.: Galois-type connections and continuities of pairs of relations. J. Int. Math. Virt. Inst. 2, 39–66 (2012)
Száz, Á.: Lower semicontinuity properties of relations in relator spaces. Adv. Stud. Contemp. Math. (Kyungshang) 23, 107–158 (2013)
Száz, Á.: Inclusions for compositions and box products of relations. J. Int. Math. Virt. Inst. 3, 97–125 (2013)
Száz, Á.: Galois and Pataki connections revisited. Tech. Rep., Inst. Math., Univ. Debrecen 2013/3, 20 pp. (2013)
Száz, Á.: A particular Galois connection between relations and set functions. Acta Univ. Sapientiae Math. 6, 73–91 (2014)
Száz, Á.: Generalizations of Galois and Pataki connections to relator spaces. J. Int. Math. Virt. Inst. 4, 43–75 (2014)
Száz, Á.: Basic tools, increasing functions, and closure operations in generalized ordered sets. In: Pardalos, P.M., Rassias, Th.M. (eds.) Contributions in Mathematics and Engineering: In Honor of Constantin Caratheodory, to appear
Száz, Á.: A unifying framework for studying continuity, increasingness, and Galois connections, submitted
Tietze, H.: Beiträge zur allgemeinen Topologie I. Axiome für verschiedene Fassungen des Umgebungsbegriffs. Math. Ann. 88, 290–312 (1923)
Weil, A.: Sur les espaces á structure uniforme et sur la topologie générale. Actual. Sci. Ind., vol. 551. Herman and Cie, Paris (1937)
Acknowledgements
The work of the first author was supported by the Hungarian Scientific Research Fund (OTKA) Grant K-111651.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Száz, Á., Zakaria, A. (2016). Mild Continuity Properties of Relations and Relators in Relator Spaces. In: Rassias, T., Pardalos, P. (eds) Essays in Mathematics and its Applications. Springer, Cham. https://doi.org/10.1007/978-3-319-31338-2_16
Download citation
DOI: https://doi.org/10.1007/978-3-319-31338-2_16
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-31336-8
Online ISBN: 978-3-319-31338-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)