Special Issues

Generalizations of some ordinary and extreme connectedness properties of topological spaces to relator spaces

  • Received: 01 December 2019 Revised: 01 February 2020
  • Primary: 54E15, 54D05; Secondary: 54G15, 54G20

  • Motivated by some ordinary and extreme connectedness properties of topologies, we introduce several reasonable connectedness properties of relators (families of relations). Moreover, we establish some intimate connections among these properties.

    More concretely, we investigate relationships among various minimalness (well-chainedness), connectedness, hyper- and ultra-connectedness, door, superset, submaximality and resolvability properties of relators.

    Since most generalized topologies and all proper stacks (ascending systems) can be derived from preorder relators, the results obtained greatly extends some standard results on topologies. Moreover, they are also closely related to some former results on well-chained and connected uniformities.

    Citation: Muwafaq Salih, Árpád Száz. Generalizations of some ordinary and extreme connectedness properties of topological spaces to relator spaces[J]. Electronic Research Archive, 2020, 28(1): 471-548. doi: 10.3934/era.2020027

    Related Papers:

  • Motivated by some ordinary and extreme connectedness properties of topologies, we introduce several reasonable connectedness properties of relators (families of relations). Moreover, we establish some intimate connections among these properties.

    More concretely, we investigate relationships among various minimalness (well-chainedness), connectedness, hyper- and ultra-connectedness, door, superset, submaximality and resolvability properties of relators.

    Since most generalized topologies and all proper stacks (ascending systems) can be derived from preorder relators, the results obtained greatly extends some standard results on topologies. Moreover, they are also closely related to some former results on well-chained and connected uniformities.



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    [1] Submaximal and spectral spaces. Math. Proc. Royal Irish Acad. (2008) 108: 137-147.
    [2] The inverse images of hyperconnected sets. Mat. Vesn. (1985) 37: 177-181.
    [3] Properties of hyperconnected spaces, their mappings into Hausdorff spaces and embeddings into hyperconnected spaces. Acta Math. Hung. (1992) 60: 41-49.
    [4] Zur Begründung der $n$-dimensionalen mengentheorischen Topologie. Math. Ann. (1925) 94: 296-308.
    [5] On connected irresolvable Hausdorff spaces. Proc. Amer. Math. Soc. (1965) 16: 463-466.
    [6] On submaximal spaces. Topology Appl. (1995) 64: 219-241.
    [7] Connection properties in nearness spaces. Canad. Math. Bull. (1985) 28: 212-217.
    [8] On uniform connecedness in nearness spaces. Math. Japonica (1995) 42: 279-282.
    [9] Door spaces on generalized topology. Int. J. Comput. Sci. Math. (2014) 6: 69-75.
    [10] Submaximal and door compactifications. Topology Appl (2011) 158: 1969-1975.
    [11] On minimal open sets and maps in topological spaces. J. Comp. Math. Sci. (2011) 2: 208-220.
    [12] On $\frak{T}$-hyperconnected spaces. Bull. Allahabad Math. Soc. (2014) 29: 15-25.
    [13] G. Birkhoff, Lattice Theory, Amer. Math. Soc. Colloq. Publ. 25, Providence, RI, 1967.
    [14] (1972) Residuation Theory.Pergamon Press.
    [15] On some variants of connectedness. Acta Math. Hungar. (1998) 79: 117-122.
    [16] Hyperconnectivity of hyperspaces. Math. Japon. (1985) 30: 757-761.
    [17] $(\omega)$topological connectedness and hyperconnectedness. Note Mat. (2011) 31: 93-101.
    [18] N. Bourbaki, General Topology, Chap 1–4, Springer-Verlag, Berlin, 1989.
    [19] N. Bourbaki, Éléments de Mathématique, Algébre Commutative, Chap. 1–4, Springer, Berlin, 2006.
    [20] A more important Galois connection between distance functions and inequality relations. Sci. Ser. A Math. Sci. (N.S.) (2009) 18: 17-38.
    [21] Über unedliche, linearen Punktmannigfaltigkeiten. Math. Ann. (1983) 21: 545-591.
    [22] E. Čech, Topological Spaces, Academia, Prague, 1966.
    [23] Debse sets, nowhere dense sets and an ideal in generalized closure spaces. Mat. Vesnik (2007) 59: 181-188.
    [24] Hyperconnectedness and extremal disconnectedness in $(\alpha)$topological spaces. Hacet. J. Math. Stat. (2015) 44: 289-294.
    [25] On uiform connection properties. Amer. Math. Monthly (1971) 78: 372-374.
    [26] Resolvability: A selective survey and some new results. Topology Appl. (1996) 74: 149-167.
    [27] (1963) Foundations of General Topology.Pergamon Press.
    [28] Á. Császár, General Topology, Adam Hilger, Bristol, 1978.
    [29] $\gamma$-connected sets. Acta Math. Hungar. (2003) 101: 273-279.
    [30] Extremally disconnected generalized topologies. Ann. Univ. Sci Budapest (2004) 47: 91-96.
    [31] Fine quasi-uniformities. Ann. Univ. Budapest (1979/1980) 22/23: 151-158.
    [32] Curtis, D. W. and Mathews, J. C., Generalized uniformities for pairs of spaces, Topology Conference, Arizona State University, Tempe, Arizona, 1967,212–246.
    [33] (2002) Introduction to Lattices and Order.Cambridge University Press.
    [34] Indexed systems of neighbordoods for general topological spaces. Amer. Math. Monthly (1961) 68: 886-894.
    [35] A counterexample on completeness in relator spaces. Publ. Math. Debrecen (1992) 41: 307-309.
    [36] K. Denecke, M. Erné and S. L. Wismath (Eds.), Galois Connections and Applications, Kluwer Academic Publisher, Dordrecht, 2004.
    [37] D. Doičinov, A unified theory of topological spaces, proximity spaces and uniform spaces, Dokl. Acad. Nauk SSSR, 156 (1964), 21–24. (Russian)
    [38] On superconnected spaces. Serdica (1994) 20: 345-350.
    [39] On door spaces. Indian J. Pure Appl. Math. (1995) 26: 873-881.
    [40] On submaximal spaces. Tamkang J. Math. (1995) 26: 243-250.
    [41] On minimal door, minimal anti-compact and minimal $T_{3/4}$–spaces. Math. Proc. Royal Irish Acad. (1998) 98: 209-215.
    [42] Ideal resolvability. Topology Appl. (1999) 93: 1-16.
    [43] Applications of maximal topologies. Top. Appl. (1993) 51: 125-139.
    [44] $F$-door spaces and $F$-submaximal spaces. Appl. Gen. Topol. (2013) 14: 97-113.
    [45] On some concepts of weak connectedness of topological spaces. Acta Math. Hungar. (2006) 110: 81-90.
    [46] V. A. Efremovič, The geometry of proximity, Mat. Sb., 31 (1952), 189–200. (Russian)
    [47] V. A. Efremović and A. S. Švarc, A new definition of uniform spaces. Metrization of proximity spaces, Dokl. Acad. Nauk. SSSR, 89 (1953), 393–396. (Russian)
    [48] Generalized hyperconnectedness. Acta Math. Hungar. (2011) 133: 140-147.
    [49] Generalized submaximal spaces. Acta Math. Hungar. (2012) 134: 132-138.
    [50] Connectedness in ideal topological spaces. Novi Sad J. Math. (2008) 38: 65-70.
    [51] *-hyperconnected ideal topological spaces. An. Stiint. Univ. Al. I. Cuza Iasi (2012) 58: 121-129.
    [52] R. Engelking, General Topology, Polish Scientific Publishers, Warszawa, 1977.
    [53] Some results on locally hyperconnected spaces. Ann. Soc. Sci. Bruxelles, Sér. I (1983) 97: 3-9.
    [54] U. V. Fattech and D. Singh, A note on $D$-spaces, Bull. Calcutta Math. Soc., 75 (1983), 363–368.
    [55] P. Fletcher and W. F. Lindgren, Quasi-Uniform Spaces, Marcel Dekker, New York, 1982.
    [56] A tale of topology. Amer. Math. Monthly (2010) 117: 663-672.
    [57] (1964) Point Set Topology.Academic Press.
    [58] Preopen sets and resolvable spaces. Kyungpook Math. J. (1987) 27: 135-143.
    [59] B. Ganter and R. Wille, Formal Concept Analysis, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-642-59830-2
    [60] Dense sets and irresolvable spaces. Ricerche Mat. (1987) 36: 163-170.
    [61] Nowhere dense sets and hyperconnected $s$-topological spaces. Bull. Cal. Math. Soc. (2000) 92: 55-58.
    [62] On parirwise hyperconnected spaces. Soochow J. Math. (2001) 27: 391-399.
    [63] On irresolvable spaces. Bull. Cal. Math. Soc. (2003) 95: 107-112.
    [64] G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove and D. S. Scott, A Compendium of Continuous Lattices, Springer-Verlag, Berlin, 1980.
    [65] Generated preorders and equivalences. Acta Acad. Paed. Agrienses, Sect. Math. (2002) 29: 95-103.
    [66] Preorders and equivalences generated by commuting relations. Acta Math. Acad. Paedagog. Nyházi. (N.S.) (2002) 18: 53-56.
    [67] Generalized hyper connected space in bigeneralized topological space. Int. J. Math. Trends Technology (2017) 47: 27-103.
    [68] A connected topology for the integers. Amer. Math. Monthly (1959) 66: 663-665.
    [69] Quasi-metrization and completion for Pervin's quasi-uniformity. Stohastica (1982) 6: 151-156.
    [70] Connected expansions of topologies. Bull. Austral. Math. Soc. (1973) 9: 259-265.
    [71] F. Hausdorff, Grundzüge der Mengenlehre, (German) Chelsea Publishing Company, New York, N. Y., 1949.
    [72] Topological Structeres. Math. Centre Tracts (1974) 52: 59-122.
    [73] A problem of set-theoretic topology. Duke Math. J (1943) 10: 309-333.
    [74] Construction of quasi-uniformities. Math. Ann. (1970) 188: 39-42.
    [75] Finite and $\omega$-resolvability. Proc. Amer. Math. Soc. (1996) 124: 1243-1246.
    [76] J. R. Isbell, Uniform Spaces, Amer. Math. Soc., Providence, 1964.
    [77] Properties of $\beta$-connected spaces. Acta Math. Hungar. (2003) 101: 227-236.
    [78] I. M. James, Topological and Uniform Structures, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-4716-6
    [79] Examples on irresolvability. Scientiae Math. Japon. (2013) 76: 461-469.
    [80] J. L. Kelley, General Topology, Van Nostrand Reinhold Company, New York, 1955.
    [81] Two theorems on relations. Trans. Amer. Math. Soc. (1963) 107: 1-9.
    [82] Computer graphics and connected topologies on finite ordered sets. Topology Appl. (1990) 36: 1-17.
    [83] H. J. Kowalsky, Topologische Räumen, Birkhäuser, Basel, 1960.
    [84] Hyperconnected type spaces. Acta Cienc. Indica Math. (2005) 31: 273-275.
    [85] (1966) Topologie I.Revised and augmented eddition: Topology I, Academic Press.
    [86] J. Kurdics, A note on connection properties, Acta Math. Acad. Paedagog. Nyházi., 12, (1990), 57–59.
    [87] J. Kurdics, Connected and Well-Chained Relator Spaces, Doctoral Dissertation, Lajos Kossuth University, Debrecen, 1991, 30 pp. (Hungarian)
    [88] Connectedness and well-chainedness properties of symmetric covering relators. Pure Math. Appl. (1991) 2: 189-197.
    [89] Connected relator spaces. Publ. Math. Debrecen (1992) 40: 155-164.
    [90] J. Kurdics and Á. Száz, Well-chained relator spaces, Kyungpook Math. J., 32 (1992), 263–271.
    [91] Well-chainedness characterizations of connected relators. Math. Pannon. (1993) 4: 37-45.
    [92] R. E. Larson, Minimum and maximum topological spaces, Bull. Acad. Polon., 18 (1970), 707–710.
    [93] On submaximal and quasi-submaximal spaces. Honam Math. J. (2010) 32: 643-649.
    [94] Stronger forms of connectivity. Rend. Circ. Mat. Palermo (1972) 21: 255-266.
    [95] Strongly connected sets in topology. Amer. Math. Monthly (1965) 72: 1098-1101.
    [96] The superset topology. Amer. Math. Monthly (1968) 75: 745-746.
    [97] Dense topologies. Amer. Math. Monthly (1968) 75: 847-852.
    [98] On uniformities generated by equivalence relations. Rend. Circ. Mat. Palermo (1969) 18: 62-70.
    [99] On Pervin's quasi uniformity. Math. J. Okayama Univ. (1970) 14: 97-102.
    [100] Well-chained uniformities. Kyungpook Math. J. (1971) 11: 143-149.
    [101] The finite square semi-uniformity. Kyungpook Math. J. (1973) 13: 179-184.
    [102] Connectedness of a stronger type in topological spaces. Nanta Math. (1979) 12: 102-109.
    [103] A note on spaces via dense sets. Tamkang J. Math. (1993) 24: 333-339.
    [104] A note on submaximal spaces and SMPC functions. Demonstratio Math. (1995) 28: 567-573.
    [105] An equation for families of relations. Pure Math. Apll., Ser. B (1990) 1: 185-188.
    [106] J. Mala, Relator Spaces, Doctoral Dissertation, Lajos Kossuth University, Debrecen, 1990, 48 pp. (Hungarian)
    [107] Relators generating the same generalized topology. Acta Math. Hungar. (1992) 60: 291-297.
    [108] Finitely generated quasi-proximities. Period. Math. Hungar. (1997) 35: 193-197.
    [109] On proximal properties of proper symmetrizations of relators. Publ. Math. Debrecen (2001) 58: 1-7.
    [110] Equations for families of relations can also be solved. C. R. Math. Rep. Acad. Sci. Canada (1990) 12: 109-112.
    [111] Properly topologically conjugated relators. Pure Math. Appl., Ser. B (1992) 3: 119-136.
    [112] Modifications of relators. Acta Math. Hungar. (1997) 77: 69-81.
    [113] Z. P. Mamuzić, Introduction to General Topology, Noordhoff, Groningen, 1963.
    [114] A note on well-chained spaces. Amer. Math. Monthly (1968) 75: 273-275.
    [115] P. M. Mathew, On hyperconnected spaces, Indian J. Pure Appl. Math., 19 (1988), 1180–1184.
    [116] On ultraconnected spaces. Int. J. Math. Sci. (1990) 13: 349-352.
    [117] A note on well-chained spaces. Amer. Math. Monthly (1968) 75: 273-275.
    [118] Door spaces are identifiable. Proc. Roy. Irish Acad. (1987) 87A: 13-16.
    [119] Relativization in resolvability and irresolvability. Int. Math. Forum (2011) 6: 1059-1064.
    [120] On uniform connectedness. Proc. Amer. Mth. Soc. (1964) 15: 446-449.
    [121] An alternative theorem for continuous relations and its applications. Publ. Inst. Math. (Beograd) (1983) 33: 163-168.
    [122] M. G. Murdeshwar and S. A. Naimpally, Quasi-Uniform Topological Spaces, Noordhoff, Groningen, 1966.
    [123] L. Nachbin, Topology and Order, D. Van Nostrand, Princeton, 1965.
    [124] (1970) Proximity Spaces.Cambridge University Press.
    [125] Connector theory. Pacific J. Math. (1975) 56: 195-213.
    [126] On $(1, 2)\alpha$-hyperconnected spaces. Int. J. Math. Anal. (2006) 3: 121-129.
    [127] On ultrapseudocompact and related spaces. Ann. Acad. Sci. Fennicae (1977) 3: 185-205.
    [128] A note on hyperconnected sets. Mat. Vesnik (1979) 3: 53-60.
    [129] Functions which preserve hyperconnected spaces. Rev. Roum. Math. Pures Appl. (1980) 25: 1091-1094.
    [130] Hyperconnectedness and preopen sets. Rev. Roum. Math. Pures Appl. (1984) 29: 329-334.
    [131] Properties of hyperconnected spaces. Acta Math. Hungar. (1995) 66: 147-154.
    [132] An example of a connected irresolvable Hausdorff space. Duke Math. J. (1953) 20: 513-520.
    [133] W. Page, Topological Uniform Structures, John Wiley and Sons Inc, New York, 1978.
    [134] G. Pataki, Supplementary notes to the theory of simple relators, Radovi Mat., 9 (1999), 101–118.
    [135] On the extensions, refinements and modifications of relators. Math. Balk. (2001) 15: 155-186.
    [136] G. Pataki, Well-chained, Connected and Simple Relators, Ph.D Dissertation, Debrecen, 2004.
    [137] A unified treatment of well-chainedness and connectedness properties. Acta Math. Acad. Paedagog. Nyházi. (N.S.) (2003) 19: 101-165.
    [138] Uniformizations of neighborhood axioms. Math. Ann. (1962) 147: 313-315.
    [139] Quasi-uniformization of topological spaces. Math. Ann. (1962) 147: 316-317.
    [140] Quasi-proximities for topological spaces. Math. Ann. (1963) 150: 325-326.
    [141] Connectedness in bitopological spaces. Indag. Math. (1967) 29: 369-372.
    [142] Spazi semiconnessi e spazi semiaperty. Rend. Circ. Mat. Palermo (1975) 24: 273-285.
    [143] Semicontinuity and closedness properties of relations in relator spaces. Mathematica (Cluj) (2003) 45: 73-92.
    [144] D. Rendi and B. Rendi, On relative $n$-connectedness, The 7th Symposium of Mathematics and Applications, "Politehnica" University of Timisoara, Romania, 1997,304–308.
    [145] On generalizations of hyperconnected spaces. J. Adv. Res. Pure Math. (2012) 4: 46-58.
    [146] Remarks on generalized hypperconnectedness. Acta Math. Hung. (2012) 136: 157-164.
    [147] Die Genesis der Raumbegriffs. Math. Naturwiss. Ber. Ungarn (1907) 24: 309-353.
    [148] Stetigkeitsbegriff und abstrakte Mengenlehre. Atti Ⅳ Congr. Intern.Mat., Roma (1908) Ⅱ: 18-24.
    [149] D. Rose, K. Sizemore and B. Thurston, Strongly irresolvable spaces, International Journal of Mathematics and Mathematical Sciences, 2006 (2006), Art. ID 53653, 12 pp. doi: 10.1155/IJMMS/2006/53653
    [150] H. M. Salih, On door spaces, Journal of College of Education, Al-Mustansnyah University, Bagdad, Iraq, 3 (2006), 112–117.
    [151] On sub-, pseeudo- and quasimaximalspaces. Comm. Math. Univ. Carolinae (1998) 39: 197-206.
    [152] Some answers concerning submaximal spaces. Questions and Answers in General Topology (1999) 17: 221-225.
    [153] On some properties of hyperconnected spaces. Mat. Vesnik (1977) 14: 25-27.
    [154] A note on generalized connectedness. Acta Math. Hungar. (2009) 122: 231-235.
    [155] Connectedness in syntopogeneous spaces. Proc. Amer. Math. Soc. (1964) 15: 590-595.
    [156] W. Sierpinski, General Topology, Mathematical Expositions, 7, University of Toronto Press, Toronto, 1952.
    [157] Yu. M. Smirnov, On proximity spaces, Math. Sb., 31 (1952), 543–574. (Russian.)
    [158] L. A. Steen and J. A. Seebach, Counterexamples in Topology, Springer-Verlag, New York, 1970.
    [159] The lattice of topologies: Structure and complementation. Trans. Amer. Math. Soc. (1966) 122: 379-398.
    [160] Defining nets for integration. Publ. Math. Debrecen (1989) 36: 237-252.
    [161] Á. Száz, Coherences instead of convergences, Proc. Conf. Convergence and Generalized Functions (Katowice, Poland, 1983), Polish Acad Sci., Warsaw, 1984,141–148.
    [162] Basic tools and mild continuities in relator spaces. Acta Math. Hungar. (1987) 50: 177-201.
    [163] Á. Száz, Directed, topological and transitive relators, Publ. Math. Debrecen, 35 (1988), 179–196.
    [164] Projective and inductive generations of relator spaces. Acta Math. Hungar. (1989) 53: 407-430.
    [165] Lebesgue relators. Monatsh. Math. (1990) 110: 315-319.
    [166] Á. Száz, The fat and dense sets are more important than the open and closed ones, Abstracts of the Seventh Prague Topological Symposium, Inst. Math. Czechoslovak Acad. Sci., 1991, p106.
    [167] Á. Száz, Relators, Nets and Integrals, Unfinished doctoral thesis, Debrecen, 1991.
    [168] Inverse and symmetric relators. Acta Math. Hungar. (1992) 60: 157-176.
    [169] Structures derivable from relators. Singularité (1992) 3: 14-30.
    [170] Á. Száz, Refinements of relators, Tech. Rep., Inst. Math., Univ. Debrecen, 76 (1993), 19 pp.
    [171] Cauchy nets and completeness in relator spaces. Colloq. Math. Soc. János Bolyai (1993) 55: 479-489.
    [172] Neighbourhood relators. Bolyai Soc. Math. Stud. (1995) 4: 449-465.
    [173] Relations refining and dividing each other. Pure Math. Appl. Ser. B (1995) 6: 385-394.
    [174] Á. Száz, Connectednesses of refined relators, Tech. Rep., Inst. Math., Univ. Debrecen, 1996, 6 pp.
    [175] Topological characterizations of relational properties. Grazer Math. Ber. (1996) 327: 37-52.
    [176] Uniformly, proximally and topologically compact relators. Math. Pannon. (1997) 8: 103-116.
    [177] An extension of Kelley's closed relation theorem to relator spaces. Filomat (2000) 14: 49-71.
    [178] Somewhat continuity in a unified framework for continuities of relations. Tatra Mt. Math. Publ. (2002) 24: 41-56.
    [179] A Galois connection between distance functions and inequality relations. Math. Bohem. (2002) 127: 437-448.
    [180] Upper and lower bounds in relator spaces. Serdica Math. J. (2003) 29: 239-270.
    [181] An extension of Baire's category theorem to relator spaces. Math. Morav. (2003) 7: 73-89.
    [182] Rare and meager sets in relator spaces. Tatra Mt. Math. Publ. (2004) 28: 75-95.
    [183] Á. Száz, Galois-type connections on power sets and their applications to relators, Tech. Rep., Inst. Math., Univ. Debrecen, 2 (2005), 38 pp.
    [184] Supremum properties of Galois–type connections. Comment. Math. Univ. Carolin. (2006) 47: 569-583.
    [185] Minimal structures, generalized topologies, and ascending systems should not be studied without generalized uniformities. Filomat (2007) 21: 87-97.
    [186] Á. Száz, Applications of fat and dense sets in the theory of additive functions, Tech. Rep., Inst. Math., Univ. Debrecen, 2007/3, 29 pp.
    [187] Galois type connections and closure operations on preordered sets. Acta Math. Univ. Comen. (2009) 78: 1-21.
    [188] Á. Száz, Applications of relations and relators in the extensions of stability theorems for homogeneous and additive functions, Aust. J. Math. Anal. Appl., 6 (2009), 66 pp.
    [189] Foundations of the theory of vector relators. Adv. Stud. Contemp. Math. (2010) 20: 139-195.
    [190] Galois-type connections and continuities of pairs of relations. J. Int. Math. Virt. Inst. (2012) 2: 39-66.
    [191] Lower semicontinuity properties of relations in relator spaces. Adv. Stud. Contemp. Math., (Kyungshang) (2013) 23: 107-158.
    [192] An extension of an additive selection theorem of Z. Gajda and R. Ger to vector relator spaces. Sci. Ser. A Math. Sci. (N.S.) (2013) 24: 33-54.
    [193] Inclusions for compositions and box products of relations. J. Int. Math. Virt. Inst. (2013) 3: 97-125.
    [194] A particular Galois connection between relations and set functions. Acta Univ. Sapientiae, Math. (2014) 6: 73-91.
    [195] Generalizations of Galois and Pataki connections to relator spaces. J. Int. Math. Virt. Inst. (2014) 4: 43-75.
    [196] Á. Száz, Remarks and problems at the conference on inequalities and applications, Hajdúszoboszló, Hungary, Tech. Rep., Inst. Math., Univ. Debrecen, 5 (2014), 12 pp.
    [197] Á. Száz, Basic tools, increasing functions, and closure operations in generalized ordered sets, In: P. M. Pardalos and Th. M. Rassias (Eds.), Contributions in Mathematics and Engineering: In Honor of Constantion Caratheodory, Springer, 2016,551–616.
    [198] A natural Galois connection between generalized norms and metrics. Acta Univ. Sapientiae Math. (2017) 9: 360-373.
    [199] Á. Száz, Four general continuity properties, for pairs of functions, relations and relators, whose particular cases could be investigated by hundreds of mathematicians, Tech. Rep., Inst. Math., Univ. Debrecen, 1 (2017), 17 pp.
    [200] Á. Száz, An answer to the question "What is the essential difference between Algebra and Topology?" of Shukur Al-aeashi, Tech. Rep., Inst. Math., Univ. Debrecen, 2 (2017), 6 pp.
    [201] Á. Száz, Contra continuity properties of relations in relator spaces, Tech. Rep., Inst. Math., Univ. Debrecen, 5 (2017), 48 pp.
    [202] The closure-interior Galois connection and its applications to relational equations and inclusions. J. Int. Math. Virt. Inst. (2018) 8: 181-224.
    [203] A unifying framework for studying continuity, increasingness, and Galois connections. MathLab J. (2018) 1: 154-173.
    [204] Á. Száz, Corelations are more powerful tools than relations, In: Th. M. Rassias (Ed.), Applications of Nonlinear Analysis, Springer Optimization and Its Applications, 134 (2018), 711-779.
    [205] Relationships between inclusions for relations and inequalities for corelations. Math. Pannon. (2018) 26: 15-31.
    [206] Galois and Pataki connections on generalized ordered sets. Earthline J. Math. Sci. (2019) 2: 283-323.
    [207] Á. Száz, Birelator spaces are natural generalizations of not only bitopological spaces, but also ideal topological spaces, In: Th. M. Rassias and P. M. Pardalos (Eds.), Mathematical Analysis and Applications, Springer Optimization and Its Applications, Springer Nature Switzerland AG, 154 (2019), 543-586.
    [208] Comparisons and compositions of Galois–type connections. Miskolc Math. Notes (2006) 7: 189-203.
    [209] Á. Száz and A. Zakaria, Mild continuity properties of relations and relators in relator spaces, In: P. M. Pardalos and Th. M. Rassias (Eds.), Essays in Mathematics and its Applications: In Honor of Vladimir Arnold, Springer, 2016,439–511.
    [210] Maximal connected topologies. J. Austral. Math. Soc. (1968) 8: 700-705.
    [211] T. Thompson, Characterizations of irreducible spaces, Kyungpook Math. J., 21 (1981), 191–194.
    [212] W. J. Thron, Topological Structures, Holt, Rinehart and Winston, New York, 1966.
    [213] Beiträge zur allgemeinen Topologie I. Axiome für verschiedene Fassungen des Umgebungsbegriffs. Math. Ann. (1923) 88: 290-312.
    [214] (1940) Convergence and Uniformity in Topology.Princeton Univ. Press.
    [215] A. Weil, Sur les Espaces á Structure Uniforme et sur la Topologie Générale, Actual. Sci. Ind. 551, Herman and Cie, Paris, 1937.
    [216] Evolution of the topological concept of connected. Amer. Math. Monthly (1978) 85: 720-726.
    [217] Connected door spaces and topological solutions of equations. Aequationes Math. (2018) 92: 1149-1161.
    [218] On a function of hyperconnected spaces. Demonstr. Math. (2007) 40: 939-950.
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