American Institute of Mathematical Sciences

Advanced
March  2020, 28(1): 471-548. doi: 10.3934/era.2020027

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

Muwafaq Salih , and  Árpád Száz

Institute of Mathematics, University of Debrecen, H-4032 Debrecen, Egyetem tér 1, Hungary

* Corresponding author: Muwafaq Salih

Dedicated to the memory of János Kurdics
who was the first to note that connectedness is a particular case of well-chainedness

Received  December 2019 Revised  February 2020 Published  March 2020

Fund Project: The work of the second author has been partially supported by the Hungarian Scientific Research Fund (OTKA) Grant K-111651

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.

Keywords: Generalized uniformities, well-chained, connected, hyperconnected, ultraconnected, submaximal and resolvable spaces.
Mathematics Subject Classification: Primary: 54E15, 54D05; Secondary: 54G15, 54G20.
Citation: Muwafaq Salih, Árpád Száz. Generalizations of some ordinary and extreme connectedness properties of topological spaces to relator spaces. Electronic Research Archive, 2020, 28 (1) : 471-548. doi: 10.3934/era.2020027
References:
[1]

M. E. Adams, K. Belaid, L. Diri and O. Echi, Submaximal and spectral spaces, Math. Proc. Royal Irish Acad., 108 (2008), 137-147.  doi: 10.3318/PRIA.2008.108.2.137.  Google Scholar

[2]

B. Ahmad and T. Noiri, The inverse images of hyperconnected sets, Mat. Vesn., 37 (1985), 177-181.   Google Scholar

[3]

N. Ajmal and J. K. Kohli, Properties of hyperconnected spaces, their mappings into Hausdorff spaces and embeddings into hyperconnected spaces, Acta Math. Hung., 60 (1992), 41-49.  doi: 10.1007/BF00051755.  Google Scholar

[4]

P. Alexandroff, Zur Begründung der $n$-dimensionalen mengentheorischen Topologie, Math. Ann., 94 (1925), 296-308.  doi: 10.1007/BF01208660.  Google Scholar

[5]

D. R. Anderson, On connected irresolvable Hausdorff spaces, Proc. Amer. Math. Soc., 16 (1965), 463-466.  doi: 10.2307/2034674.  Google Scholar

[6]

A. V. Arhangelskij and P. J. Collins, On submaximal spaces, Topology Appl., 64 (1995), 219-241.  doi: 10.1016/0166-8641(94)00093-I.  Google Scholar

[7]

D. Baboolal, H. L. Bentley and R. G. Ori, Connection properties in nearness spaces, Canad. Math. Bull., 28 (1985), 212-217.  doi: 10.4153/CMB-1985-024-5.  Google Scholar

[8]

D. Baboolal and R. G. Ori, On uniform connecedness in nearness spaces, Math. Japonica, 42 (1995), 279-282.   Google Scholar

[9]

A. P. D. Balan and R. M. S. Sundary, Door spaces on generalized topology, Int. J. Comput. Sci. Math., 6 (2014), 69-75.   Google Scholar

[10]

K. Belaid, L. Dridi and O. Echi, Submaximal and door compactifications, Topology Appl, 158 (2011), 1969-1975.  doi: 10.1016/j.topol.2011.06.039.  Google Scholar

[11]

S. S. Benchalli, B. M. Ittanagi and R. S. Wali, On minimal open sets and maps in topological spaces, J. Comp. Math. Sci., 2 (2011), 208-220.   Google Scholar

[12]

K. Bhavani and D. Sivaraj, On $\frak{T}$-hyperconnected spaces, Bull. Allahabad Math. Soc., 29 (2014), 15-25.   Google Scholar

[13]

G. Birkhoff, Lattice Theory, Amer. Math. Soc. Colloq. Publ. 25, Providence, RI, 1967. Google Scholar

[14] T. S. Blyth and M. F. Janowitz, Residuation Theory, Pergamon Press, Oxford, 1972.   Google Scholar
[15]

M. Bognár, On some variants of connectedness, Acta Math. Hungar., 79 (1998), 117-122.  doi: 10.1023/A:1006561806310.  Google Scholar

[16]

C. R. Borges, Hyperconnectivity of hyperspaces, Math. Japon., 30 (1985), 757-761.   Google Scholar

[17]

M. K. Bose and R. Tiwari, $(\omega)$topological connectedness and hyperconnectedness, Note Mat., 31 (2011), 93-101.   Google Scholar

[18]

N. Bourbaki, General Topology, Chap 1–4, Springer-Verlag, Berlin, 1989.  Google Scholar

[19]

N. Bourbaki, Éléments de Mathématique, Algébre Commutative, Chap. 1–4, Springer, Berlin, 2006. Google Scholar

[20]

S. Buglyó and Á. Száz, A more important Galois connection between distance functions and inequality relations, Sci. Ser. A Math. Sci. (N.S.), 18 (2009), 17-38.   Google Scholar

[21]

G. Cantor, Über unedliche, linearen Punktmannigfaltigkeiten, Math. Ann., 21 (1983), 545-591.  doi: 10.1007/BF01446819.  Google Scholar

[22]

E. Čech, Topological Spaces, Academia, Prague, 1966. Google Scholar

[23]

C. Chattopadhyay, Debse sets, nowhere dense sets and an ideal in generalized closure spaces, Mat. Vesnik, 59 (2007), 181-188.   Google Scholar

[24]

A. R. Choudhury, A. Mukharjee and M. K. Bose, Hyperconnectedness and extremal disconnectedness in $(\alpha)$topological spaces, Hacet. J. Math. Stat., 44 (2015), 289-294.   Google Scholar

[25]

P. J. Collins, On uiform connection properties, Amer. Math. Monthly, 78 (1971), 372-374.  doi: 10.1080/00029890.1971.11992762.  Google Scholar

[26]

W. W. Comfort and S. Garcia-Ferreira, Resolvability: A selective survey and some new results, Topology Appl., 74 (1996), 149-167.  doi: 10.1016/S0166-8641(96)00052-1.  Google Scholar

[27] Á. Császár, Foundations of General Topology, Pergamon Press, London, 1963.   Google Scholar
[28]

Á. Császár, General Topology, Adam Hilger, Bristol, 1978.  Google Scholar

[29]

Á. Császár, $\gamma$-connected sets, Acta Math. Hungar., 101 (2003), 273-279.  doi: 10.1023/B:AMHU.0000004939.57085.9e.  Google Scholar

[30]

Á. Császár, Extremally disconnected generalized topologies, Ann. Univ. Sci Budapest, 47 (2004), 91-96.   Google Scholar

[31]

Á. Császár and R. Z. Domiaty, Fine quasi-uniformities, Ann. Univ. Budapest, 22/23 (1979/1980), 151-158.   Google Scholar

[32]

Curtis, D. W. and Mathews, J. C., Generalized uniformities for pairs of spaces, Topology Conference, Arizona State University, Tempe, Arizona, 1967,212–246.  Google Scholar

[33] B. A. Davey and H. A. Priestley, Introduction to Lattices and Order, Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511809088.  Google Scholar
[34]

A. S. Davis, Indexed systems of neighbordoods for general topological spaces, Amer. Math. Monthly, 68 (1961), 886-894.  doi: 10.1080/00029890.1961.11989785.  Google Scholar

[35]

J. Deák, A counterexample on completeness in relator spaces, Publ. Math. Debrecen, 41 (1992), 307-309.   Google Scholar

[36]

K. Denecke, M. Erné and S. L. Wismath (Eds.), Galois Connections and Applications, Kluwer Academic Publisher, Dordrecht, 2004. Google Scholar

[37]

D. Doičinov, A unified theory of topological spaces, proximity spaces and uniform spaces, Dokl. Acad. Nauk SSSR, 156 (1964), 21–24. (Russian)  Google Scholar

[38]

J. Dontchev, On superconnected spaces, Serdica, 20 (1994), 345-350.   Google Scholar

[39]

J. Dontchev, On door spaces, Indian J. Pure Appl. Math., 26 (1995), 873-881.   Google Scholar

[40]

J. Dontchev, On submaximal spaces, Tamkang J. Math., 26 (1995), 243-250.   Google Scholar

[41]

J. Dontchev, M. Ganster, G. J. Kennedy and S. D. McCartan, On minimal door, minimal anti-compact and minimal $T_{3/4}$–spaces, Math. Proc. Royal Irish Acad., 98 (1998), 209-215.   Google Scholar

[42]

J. Dontchev, M. Ganster and D. Rose, Ideal resolvability, Topology Appl., 93 (1999), 1-16.  doi: 10.1016/S0166-8641(97)00257-5.  Google Scholar

[43]

E. K. van Douwen, Applications of maximal topologies, Top. Appl., 51 (1993), 125-139.  doi: 10.1016/0166-8641(93)90145-4.  Google Scholar

[44]

L. Dridi, S. Lazaar and T. Turki, $F$-door spaces and $F$-submaximal spaces, Appl. Gen. Topol., 14 (2013), 97-113.  doi: 10.4995/agt.2013.1621.  Google Scholar

[45]

Z. Duszyński, On some concepts of weak connectedness of topological spaces, Acta Math. Hungar., 110 (2006), 81-90.  doi: 10.1007/s10474-006-0008-x.  Google Scholar

[46]

V. A. Efremovič, The geometry of proximity, Mat. Sb., 31 (1952), 189–200. (Russian)  Google Scholar

[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)  Google Scholar

[48]

E. Ekici, Generalized hyperconnectedness, Acta Math. Hungar., 133 (2011), 140-147.  doi: 10.1007/s10474-011-0086-2.  Google Scholar

[49]

E. Ekici, Generalized submaximal spaces, Acta Math. Hungar., 134 (2012), 132-138.  doi: 10.1007/s10474-011-0109-z.  Google Scholar

[50]

E. Ekici and T. Noiri, Connectedness in ideal topological spaces, Novi Sad J. Math., 38 (2008), 65-70.   Google Scholar

[51]

E. Ekici and T. Noiri, *-hyperconnected ideal topological spaces, An. Stiint. Univ. Al. I. Cuza Iasi, 58 (2012), 121-129.   Google Scholar

[52]

R. Engelking, General Topology, Polish Scientific Publishers, Warszawa, 1977.  Google Scholar

[53]

U. V. Fattech and D. Singh, Some results on locally hyperconnected spaces, Ann. Soc. Sci. Bruxelles, Sér. I, 97 (1983), 3-9.   Google Scholar

[54]

U. V. Fattech and D. Singh, A note on $D$-spaces, Bull. Calcutta Math. Soc., 75 (1983), 363–368.  Google Scholar

[55]

P. Fletcher and W. F. Lindgren, Quasi-Uniform Spaces, Marcel Dekker, New York, 1982.  Google Scholar

[56]

G. B. Folland, A tale of topology, Amer. Math. Monthly, 117 (2010), 663-672.  doi: 10.4169/000298910x515730.  Google Scholar

[57] S. A. Gaal, Point Set Topology, Academic Press, New York, 1964.   Google Scholar
[58]

M. Ganster, Preopen sets and resolvable spaces, Kyungpook Math. J., 27 (1987), 135-143.   Google Scholar

[59]

B. Ganter and R. Wille, Formal Concept Analysis, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-642-59830-2.  Google Scholar

[60]

M. Ganster, I. L. Reilly and M. K. Vamanamurthy, Dense sets and irresolvable spaces, Ricerche Mat., 36 (1987), 163-170.   Google Scholar

[61]

B. Garai and C. Bandyopadhyay, Nowhere dense sets and hyperconnected $s$-topological spaces, Bull. Cal. Math. Soc., 92 (2000), 55-58.   Google Scholar

[62]

B. Garai and C. Bandyopadhyay, On parirwise hyperconnected spaces, Soochow J. Math., 27 (2001), 391-399.   Google Scholar

[63]

B. Garai and C. Bandyopadhyay, On irresolvable spaces, Bull. Cal. Math. Soc., 95 (2003), 107-112.   Google Scholar

[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.  Google Scholar

[65]

T. Glavosits, Generated preorders and equivalences, Acta Acad. Paed. Agrienses, Sect. Math., 29 (2002), 95-103.   Google Scholar

[66]

T. Glavosits, Preorders and equivalences generated by commuting relations, Acta Math. Acad. Paedagog. Nyházi. (N.S.), 18 (2002), 53-56.   Google Scholar

[67]

R. G. D. Gnanam, Generalized hyper connected space in bigeneralized topological space, Int. J. Math. Trends Technology, 47 (2017), 27-103.   Google Scholar

[68]

S. W. Golomb, A connected topology for the integers, Amer. Math. Monthly, 66 (1959), 663-665.  doi: 10.1080/00029890.1959.11989385.  Google Scholar

[69]

V. Gregori and J. Ferrer, Quasi-metrization and completion for Pervin's quasi-uniformity, Stohastica, 6 (1982), 151-156.   Google Scholar

[70]

J. A. Guthrie, D. F. Reynolds and and H. E. Stone, Connected expansions of topologies, Bull. Austral. Math. Soc., 9 (1973), 259-265.  doi: 10.1017/S000497270004315X.  Google Scholar

[71]

F. Hausdorff, Grundzüge der Mengenlehre, (German) Chelsea Publishing Company, New York, N. Y., 1949.  Google Scholar

[72]

H. Herrlich, Topological Structeres, Math. Centre Tracts, 52 (1974), 59-122.   Google Scholar

[73]

E. Hewitt, A problem of set-theoretic topology, Duke Math. J, 10 (1943), 309-333.  doi: 10.1215/S0012-7094-43-01029-4.  Google Scholar

[74]

W. Hunsaker and W. Lindgren, Construction of quasi-uniformities, Math. Ann., 188 (1970), 39-42.  doi: 10.1007/BF01435413.  Google Scholar

[75]

A. Illanes, Finite and $\omega$-resolvability, Proc. Amer. Math. Soc., 124 (1996), 1243-1246.  doi: 10.1090/S0002-9939-96-03348-5.  Google Scholar

[76]

J. R. Isbell, Uniform Spaces, Amer. Math. Soc., Providence, 1964. Google Scholar

[77]

S. Jafari and T. Noiri, Properties of $\beta$-connected spaces, Acta Math. Hungar., 101 (2003), 227-236.  doi: 10.1023/B:AMHU.0000003907.90823.79.  Google Scholar

[78]

I. M. James, Topological and Uniform Structures, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-4716-6.  Google Scholar

[79]

A. Kalapodi, Examples on irresolvability, Scientiae Math. Japon., 76 (2013), 461-469.   Google Scholar

[80]

J. L. Kelley, General Topology, Van Nostrand Reinhold Company, New York, 1955.  Google Scholar

[81]

H. Kenyon, Two theorems on relations, Trans. Amer. Math. Soc., 107 (1963), 1-9.  doi: 10.1090/S0002-9947-1963-0148590-2.  Google Scholar

[82]

E. Khalimsky, R. Kopperman and P. R. Meyer, Computer graphics and connected topologies on finite ordered sets, Topology Appl., 36 (1990), 1-17.  doi: 10.1016/0166-8641(90)90031-V.  Google Scholar

[83]

H. J. Kowalsky, Topologische Räumen, Birkhäuser, Basel, 1960. Google Scholar

[84]

M. K. R. S. V. Kumar, Hyperconnected type spaces, Acta Cienc. Indica Math., 31 (2005), 273-275.   Google Scholar

[85] K. Kuratowski, Topologie I, Revised and augmented eddition: Topology I, Academic Press, New York, 1966.   Google Scholar
[86]

J. Kurdics, A note on connection properties, Acta Math. Acad. Paedagog. Nyházi., 12, (1990), 57–59. Google Scholar

[87]

J. Kurdics, Connected and Well-Chained Relator Spaces, Doctoral Dissertation, Lajos Kossuth University, Debrecen, 1991, 30 pp. (Hungarian) Google Scholar

[88]

J. Kurdics, J. Mala and Á. Száz, Connectedness and well-chainedness properties of symmetric covering relators, Pure Math. Appl., 2 (1991), 189-197.   Google Scholar

[89]

J. Kurdics and Á. Száz, Connected relator spaces, Publ. Math. Debrecen, 40 (1992), 155-164.   Google Scholar

[90]

J. Kurdics and Á. Száz, Well-chained relator spaces, Kyungpook Math. J., 32 (1992), 263–271.  Google Scholar

[91]

J. Kurdics and Á. Száz, Well-chainedness characterizations of connected relators, Math. Pannon., 4 (1993), 37-45.   Google Scholar

[92]

R. E. Larson, Minimum and maximum topological spaces, Bull. Acad. Polon., 18 (1970), 707–710.  Google Scholar

[93]

S. W. Lee, M. A. Moon and M. H. Cho, On submaximal and quasi-submaximal spaces, Honam Math. J., 32 (2010), 643-649.  doi: 10.5831/HMJ.2010.32.4.643.  Google Scholar

[94]

J. E. Leuschen and B. T. Sims, Stronger forms of connectivity, Rend. Circ. Mat. Palermo, 21 (1972), 255-266.  doi: 10.1007/BF02843790.  Google Scholar

[95]

N. Levine, Strongly connected sets in topology, Amer. Math. Monthly, 72 (1965), 1098-1101.  doi: 10.2307/2315958.  Google Scholar

[96]

N. Levine, The superset topology, Amer. Math. Monthly, 75 (1968), 745-746.   Google Scholar

[97]

N. Levine, Dense topologies, Amer. Math. Monthly, 75 (1968), 847-852.  doi: 10.1080/00029890.1968.11971077.  Google Scholar

[98]

N. Levine, On uniformities generated by equivalence relations, Rend. Circ. Mat. Palermo, 18 (1969), 62-70.  doi: 10.1007/BF02888946.  Google Scholar

[99]

N. Levine, On Pervin's quasi uniformity, Math. J. Okayama Univ., 14 (1970), 97-102.   Google Scholar

[100]

N. Levine, Well-chained uniformities, Kyungpook Math. J., 11 (1971), 143-149.   Google Scholar

[101]

N. Levine, The finite square semi-uniformity, Kyungpook Math. J., 13 (1973), 179-184.   Google Scholar

[102]

S. N. Maheswari and U. Tapi, Connectedness of a stronger type in topological spaces, Nanta Math., 12 (1979), 102-109.   Google Scholar

[103]

R. A. Mahmoud and D. A. Rose, A note on spaces via dense sets, Tamkang J. Math., 24 (1993), 333-339.   Google Scholar

[104]

R. A. Mahmoud and D. A. Rose, A note on submaximal spaces and SMPC functions, Demonstratio Math., 28 (1995), 567-573.   Google Scholar

[105]

J. Mala, An equation for families of relations, Pure Math. Apll., Ser. B, 1 (1990), 185-188.   Google Scholar

[106]

J. Mala, Relator Spaces, Doctoral Dissertation, Lajos Kossuth University, Debrecen, 1990, 48 pp. (Hungarian) Google Scholar

[107]

J. Mala, Relators generating the same generalized topology, Acta Math. Hungar., 60 (1992), 291-297.  doi: 10.1007/BF00051647.  Google Scholar

[108]

J. Mala, Finitely generated quasi-proximities, Period. Math. Hungar., 35 (1997), 193-197.  doi: 10.1023/A:1004553417107.  Google Scholar

[109]

J. Mala, On proximal properties of proper symmetrizations of relators, Publ. Math. Debrecen, 58 (2001), 1-7.   Google Scholar

[110]

J. Mala and Á. Száz, Equations for families of relations can also be solved, C. R. Math. Rep. Acad. Sci. Canada, 12 (1990), 109-112.   Google Scholar

[111]

J. Mala and Á. Száz, Properly topologically conjugated relators, Pure Math. Appl., Ser. B, 3 (1992), 119-136.   Google Scholar

[112]

J. Mala and Á. Száz, Modifications of relators, Acta Math. Hungar., 77 (1997), 69-81.  doi: 10.1023/A:1006583622770.  Google Scholar

[113]

Z. P. Mamuzić, Introduction to General Topology, Noordhoff, Groningen, 1963.  Google Scholar

[114]

J. C. Mathews, A note on well-chained spaces, Amer. Math. Monthly, 75 (1968), 273-275.  doi: 10.2307/2314959.  Google Scholar

[115]

P. M. Mathew, On hyperconnected spaces, Indian J. Pure Appl. Math., 19 (1988), 1180–1184.  Google Scholar

[116]

P. M. Mathew, On ultraconnected spaces, Int. J. Math. Sci., 13 (1990), 349-352.  doi: 10.1155/S0161171290000515.  Google Scholar

[117]

J. C. Mathews, A note on well-chained spaces, Amer. Math. Monthly, 75 (1968), 273-275.  doi: 10.2307/2314959.  Google Scholar

[118]

S. D. McCartan, Door spaces are identifiable, Proc. Roy. Irish Acad., 87A (1987), 13-16.   Google Scholar

[119]

S. Modak, Relativization in resolvability and irresolvability, Int. Math. Forum, 6 (2011), 1059-1064.   Google Scholar

[120]

S. G. Mrówka and W. J. Pervin, On uniform connectedness, Proc. Amer. Mth. Soc., 15 (1964), 446-449.  doi: 10.1090/S0002-9939-1964-0161307-7.  Google Scholar

[121]

Á. Münnich and Á. Száz, An alternative theorem for continuous relations and its applications, Publ. Inst. Math. (Beograd), 33 (1983), 163-168.   Google Scholar

[122]

M. G. Murdeshwar and S. A. Naimpally, Quasi-Uniform Topological Spaces, Noordhoff, Groningen, 1966.  Google Scholar

[123]

L. Nachbin, Topology and Order, D. Van Nostrand, Princeton, 1965.  Google Scholar

[124] S. A. Naimpally and B. D. Warrack, Proximity Spaces, Cambridge University Press, Cambridge, 1970.   Google Scholar
[125]

H. Nakano and K. Nakano, Connector theory, Pacific J. Math., 56 (1975), 195-213.  doi: 10.2140/pjm.1975.56.195.  Google Scholar

[126]

G. Navalagi, M. L. Thivagar, R. R. Rajeswari and S. A. Ponmani, On $(1, 2)\alpha$-hyperconnected spaces, Int. J. Math. Anal., 3 (2006), 121-129.   Google Scholar

[127]

T. Nieminen, On ultrapseudocompact and related spaces, Ann. Acad. Sci. Fennicae, 3 (1977), 185-205.  doi: 10.5186/aasfm.1977.0321.  Google Scholar

[128]

T. Noiri, A note on hyperconnected sets, Mat. Vesnik, 3 (1979), 53-60.   Google Scholar

[129]

T. Noiri, Functions which preserve hyperconnected spaces, Rev. Roum. Math. Pures Appl., 25 (1980), 1091-1094.   Google Scholar

[130]

T. Noiri, Hyperconnectedness and preopen sets, Rev. Roum. Math. Pures Appl., 29 (1984), 329-334.   Google Scholar

[131]

T. Noiri, Properties of hyperconnected spaces, Acta Math. Hungar., 66 (1995), 147-154.  doi: 10.1007/BF01874359.  Google Scholar

[132]

K. Padmavally, An example of a connected irresolvable Hausdorff space, Duke Math. J., 20 (1953), 513-520.  doi: 10.1215/S0012-7094-53-02050-X.  Google Scholar

[133]

W. Page, Topological Uniform Structures, John Wiley and Sons Inc, New York, 1978.  Google Scholar

[134]

G. Pataki, Supplementary notes to the theory of simple relators, Radovi Mat., 9 (1999), 101–118.  Google Scholar

[135]

G. Pataki, On the extensions, refinements and modifications of relators, Math. Balk., 15 (2001), 155-186.   Google Scholar

[136]

G. Pataki, Well-chained, Connected and Simple Relators, Ph.D Dissertation, Debrecen, 2004. Google Scholar

[137]

G. Pataki and Á. Száz, A unified treatment of well-chainedness and connectedness properties, Acta Math. Acad. Paedagog. Nyházi. (N.S.), 19 (2003), 101-165.   Google Scholar

[138]

W. J. Pervin, Uniformizations of neighborhood axioms, Math. Ann., 147 (1962), 313-315.  doi: 10.1007/BF01440952.  Google Scholar

[139]

W. J. Pervin, Quasi-uniformization of topological spaces, Math. Ann., 147 (1962), 316-317.  doi: 10.1007/BF01440953.  Google Scholar

[140]

W. J. Pervin, Quasi-proximities for topological spaces, Math. Ann., 150 (1963), 325-326.  doi: 10.1007/BF01470761.  Google Scholar

[141]

W. J. Pervin, Connectedness in bitopological spaces, Indag. Math., 29 (1967), 369-372.   Google Scholar

[142]

V. Pipitone and G. Russo, Spazi semiconnessi e spazi semiaperty, Rend. Circ. Mat. Palermo, 24 (1975), 273-285.  doi: 10.1007/BF02843735.  Google Scholar

[143]

Cs. Rakaczki and Á. Száz, Semicontinuity and closedness properties of relations in relator spaces, Mathematica (Cluj), 45 (2003), 73-92.   Google Scholar

[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. Google Scholar

[145]

V. Renukadevi, On generalizations of hyperconnected spaces, J. Adv. Res. Pure Math., 4 (2012), 46-58.   Google Scholar

[146]

V. Renukadevi, Remarks on generalized hypperconnectedness, Acta Math. Hung., 136 (2012), 157-164.  doi: 10.1007/s10474-011-0192-1.  Google Scholar

[147]

F. Riesz, Die Genesis der Raumbegriffs, Math. Naturwiss. Ber. Ungarn, 24 (1907), 309-353.   Google Scholar

[148]

F. Riesz, Stetigkeitsbegriff und abstrakte Mengenlehre, Atti Ⅳ Congr. Intern.Mat., Roma, Ⅱ (1908), 18-24.   Google Scholar

[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.  Google Scholar

[150]

H. M. Salih, On door spaces, Journal of College of Education, Al-Mustansnyah University, Bagdad, Iraq, 3 (2006), 112–117. Google Scholar

[151]

J. Schröder, On sub-, pseeudo- and quasimaximalspaces, Comm. Math. Univ. Carolinae, 39 (1998), 197-206.   Google Scholar

[152]

J. Schröder, Some answers concerning submaximal spaces, Questions and Answers in General Topology, 17 (1999), 221-225.   Google Scholar

[153]

A. K. Sharma, On some properties of hyperconnected spaces, Mat. Vesnik, 14 (1977), 25-27.   Google Scholar

[154]

R.-X. Shen, A note on generalized connectedness, Acta Math. Hungar., 122 (2009), 231-235.  doi: 10.1007/s10474-008-8009-6.  Google Scholar

[155]

J. L. Sieber and W. J. Pervin, Connectedness in syntopogeneous spaces, Proc. Amer. Math. Soc., 15 (1964), 590-595.  doi: 10.1090/S0002-9939-1964-0166747-8.  Google Scholar

[156]

W. Sierpinski, General Topology, Mathematical Expositions, 7, University of Toronto Press, Toronto, 1952.  Google Scholar

[157]

Yu. M. Smirnov, On proximity spaces, Math. Sb., 31 (1952), 543–574. (Russian.)  Google Scholar

[158]

L. A. Steen and J. A. Seebach, Counterexamples in Topology, Springer-Verlag, New York, 1970.  Google Scholar

[159]

A. K. Steiner, The lattice of topologies: Structure and complementation, Trans. Amer. Math. Soc., 122 (1966), 379-398.  doi: 10.1090/S0002-9947-1966-0190893-2.  Google Scholar

[160]

Gy. Szabó and Á. Száz, Defining nets for integration, Publ. Math. Debrecen, 36 (1989), 237-252.   Google Scholar

[161]

Á. Száz, Coherences instead of convergences, Proc. Conf. Convergence and Generalized Functions (Katowice, Poland, 1983), Polish Acad Sci., Warsaw, 1984,141–148. Google Scholar

[162]

Á. Száz, Basic tools and mild continuities in relator spaces, Acta Math. Hungar., 50 (1987), 177-201.  doi: 10.1007/BF01903935.  Google Scholar

[163]

Á. Száz, Directed, topological and transitive relators, Publ. Math. Debrecen, 35 (1988), 179–196.  Google Scholar

[164]

Á. Száz, Projective and inductive generations of relator spaces, Acta Math. Hungar., 53 (1989), 407-430.  doi: 10.1007/BF01953378.  Google Scholar

[165]

Á. Száz, Lebesgue relators, Monatsh. Math., 110 (1990), 315-319.  doi: 10.1007/BF01301684.  Google Scholar

[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. Google Scholar

[167]

Á. Száz, Relators, Nets and Integrals, Unfinished doctoral thesis, Debrecen, 1991. Google Scholar

[168]

Á. Száz, Inverse and symmetric relators, Acta Math. Hungar., 60 (1992), 157-176.  doi: 10.1007/BF00051766.  Google Scholar

[169]

Á. Száz, Structures derivable from relators, Singularité, 3 (1992), 14-30.   Google Scholar

[170]

Á. Száz, Refinements of relators, Tech. Rep., Inst. Math., Univ. Debrecen, 76 (1993), 19 pp. Google Scholar

[171]

Á. Száz, Cauchy nets and completeness in relator spaces, Colloq. Math. Soc. János Bolyai, 55 (1993), 479-489.   Google Scholar

[172]

Á. Száz, Neighbourhood relators, Bolyai Soc. Math. Stud., 4 (1995), 449-465.   Google Scholar

[173]

Á. Száz, Relations refining and dividing each other, Pure Math. Appl. Ser. B, 6 (1995), 385-394.   Google Scholar

[174]

Á. Száz, Connectednesses of refined relators, Tech. Rep., Inst. Math., Univ. Debrecen, 1996, 6 pp. Google Scholar

[175]

Á. Száz, Topological characterizations of relational properties, Grazer Math. Ber., 327 (1996), 37-52.   Google Scholar

[176]

Á. Száz, Uniformly, proximally and topologically compact relators, Math. Pannon., 8 (1997), 103-116.   Google Scholar

[177]

Á. Száz, An extension of Kelley's closed relation theorem to relator spaces, Filomat, 14 (2000), 49-71.   Google Scholar

[178]

Á. Száz, Somewhat continuity in a unified framework for continuities of relations, Tatra Mt. Math. Publ., 24 (2002), 41-56.   Google Scholar

[179]

Á. Száz, A Galois connection between distance functions and inequality relations, Math. Bohem., 127 (2002), 437-448.   Google Scholar

[180]

Á. Száz, Upper and lower bounds in relator spaces, Serdica Math. J., 29 (2003), 239-270.   Google Scholar

[181]

Á. Száz, An extension of Baire's category theorem to relator spaces, Math. Morav., 7 (2003), 73-89.  doi: 10.5937/matmor0307073s.  Google Scholar

[182]

Á. Száz, Rare and meager sets in relator spaces, Tatra Mt. Math. Publ., 28 (2004), 75-95.   Google Scholar

[183]

Á. Száz, Galois-type connections on power sets and their applications to relators, Tech. Rep., Inst. Math., Univ. Debrecen, 2 (2005), 38 pp. Google Scholar

[184]

Á. Száz, Supremum properties of Galois–type connections, Comment. Math. Univ. Carolin., 47 (2006), 569-583.   Google Scholar

[185]

Á. Száz, Minimal structures, generalized topologies, and ascending systems should not be studied without generalized uniformities, Filomat, 21 (2007), 87-97.  doi: 10.2298/FIL0701087S.  Google Scholar

[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. Google Scholar

[187]

Á. Száz, Galois type connections and closure operations on preordered sets, Acta Math. Univ. Comen., 78 (2009), 1-21.   Google Scholar

[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.  Google Scholar

[189]

Á. Száz, Foundations of the theory of vector relators, Adv. Stud. Contemp. Math., 20 (2010), 139-195.   Google Scholar

[190]

Á. Száz, Galois-type connections and continuities of pairs of relations, J. Int. Math. Virt. Inst., 2 (2012), 39-66.   Google Scholar

[191]

Á. Száz, Lower semicontinuity properties of relations in relator spaces, Adv. Stud. Contemp. Math., (Kyungshang), 23 (2013), 107-158.   Google Scholar

[192]

Á. Száz, An extension of an additive selection theorem of Z. Gajda and R. Ger to vector relator spaces, Sci. Ser. A Math. Sci. (N.S.), 24 (2013), 33-54.   Google Scholar

[193]

Á. Száz, Inclusions for compositions and box products of relations, J. Int. Math. Virt. Inst., 3 (2013), 97-125.   Google Scholar

[194]

Á. Száz, A particular Galois connection between relations and set functions, Acta Univ. Sapientiae, Math., 6 (2014), 73-91.  doi: 10.2478/ausm-2014-0019.  Google Scholar

[195]

Á. Száz, Generalizations of Galois and Pataki connections to relator spaces, J. Int. Math. Virt. Inst., 4 (2014), 43-75.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[198]

Á. Száz, A natural Galois connection between generalized norms and metrics, Acta Univ. Sapientiae Math., 9 (2017), 360-373.  doi: 10.1515/ausm-2017-0027.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[201]

Á. Száz, Contra continuity properties of relations in relator spaces, Tech. Rep., Inst. Math., Univ. Debrecen, 5 (2017), 48 pp. Google Scholar

[202]

Á. Száz, The closure-interior Galois connection and its applications to relational equations and inclusions, J. Int. Math. Virt. Inst., 8 (2018), 181-224.   Google Scholar

[203]

Á. Száz, A unifying framework for studying continuity, increasingness, and Galois connections, MathLab J., 1 (2018), 154-173.   Google Scholar

[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.  Google Scholar

[205]

Á. Száz, Relationships between inclusions for relations and inequalities for corelations, Math. Pannon., 26 (2018), 15-31.   Google Scholar

[206]

Á. Száz, Galois and Pataki connections on generalized ordered sets, Earthline J. Math. Sci., 2 (2019), 283-323.   Google Scholar

[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. Google Scholar

[208]

Á. Száz and J. Túri, Comparisons and compositions of Galois–type connections, Miskolc Math. Notes, 7 (2006), 189-203.  doi: 10.18514/MMN.2006.144.  Google Scholar

[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.  Google Scholar

[210]

J. P. Thomas, Maximal connected topologies, J. Austral. Math. Soc., 8 (1968), 700-705.  doi: 10.1017/S1446788700006510.  Google Scholar

[211]

T. Thompson, Characterizations of irreducible spaces, Kyungpook Math. J., 21 (1981), 191–194.  Google Scholar

[212]

W. J. Thron, Topological Structures, Holt, Rinehart and Winston, New York, 1966.  Google Scholar

[213]

H. Tietze, Beiträge zur allgemeinen Topologie I. Axiome für verschiedene Fassungen des Umgebungsbegriffs, Math. Ann., 88 (1923), 290-312.  doi: 10.1007/BF01579182.  Google Scholar

[214] J. W. Tukey, Convergence and Uniformity in Topology, Princeton Univ. Press, Princeton, 1940.   Google Scholar
[215]

A. Weil, Sur les Espaces á Structure Uniforme et sur la Topologie Générale, Actual. Sci. Ind. 551, Herman and Cie, Paris, 1937. Google Scholar

[216]

R. L. Wilder, Evolution of the topological concept of connected, Amer. Math. Monthly, 85 (1978), 720-726.  doi: 10.2307/2321676.  Google Scholar

[217]

J. Wu, C. Wang and D. Zhang, Connected door spaces and topological solutions of equations, Aequationes Math., 92 (2018), 1149-1161.  doi: 10.1007/s00010-018-0577-0.  Google Scholar

[218]

I. Zorlutuna, On a function of hyperconnected spaces, Demonstr. Math., 40 (2007), 939-950.   Google Scholar

show all references

Dedicated to the memory of János Kurdics
who was the first to note that connectedness is a particular case of well-chainedness

References:
[1]

M. E. Adams, K. Belaid, L. Diri and O. Echi, Submaximal and spectral spaces, Math. Proc. Royal Irish Acad., 108 (2008), 137-147.  doi: 10.3318/PRIA.2008.108.2.137.  Google Scholar

[2]

B. Ahmad and T. Noiri, The inverse images of hyperconnected sets, Mat. Vesn., 37 (1985), 177-181.   Google Scholar

[3]

N. Ajmal and J. K. Kohli, Properties of hyperconnected spaces, their mappings into Hausdorff spaces and embeddings into hyperconnected spaces, Acta Math. Hung., 60 (1992), 41-49.  doi: 10.1007/BF00051755.  Google Scholar

[4]

P. Alexandroff, Zur Begründung der $n$-dimensionalen mengentheorischen Topologie, Math. Ann., 94 (1925), 296-308.  doi: 10.1007/BF01208660.  Google Scholar

[5]

D. R. Anderson, On connected irresolvable Hausdorff spaces, Proc. Amer. Math. Soc., 16 (1965), 463-466.  doi: 10.2307/2034674.  Google Scholar

[6]

A. V. Arhangelskij and P. J. Collins, On submaximal spaces, Topology Appl., 64 (1995), 219-241.  doi: 10.1016/0166-8641(94)00093-I.  Google Scholar

[7]

D. Baboolal, H. L. Bentley and R. G. Ori, Connection properties in nearness spaces, Canad. Math. Bull., 28 (1985), 212-217.  doi: 10.4153/CMB-1985-024-5.  Google Scholar

[8]

D. Baboolal and R. G. Ori, On uniform connecedness in nearness spaces, Math. Japonica, 42 (1995), 279-282.   Google Scholar

[9]

A. P. D. Balan and R. M. S. Sundary, Door spaces on generalized topology, Int. J. Comput. Sci. Math., 6 (2014), 69-75.   Google Scholar

[10]

K. Belaid, L. Dridi and O. Echi, Submaximal and door compactifications, Topology Appl, 158 (2011), 1969-1975.  doi: 10.1016/j.topol.2011.06.039.  Google Scholar

[11]

S. S. Benchalli, B. M. Ittanagi and R. S. Wali, On minimal open sets and maps in topological spaces, J. Comp. Math. Sci., 2 (2011), 208-220.   Google Scholar

[12]

K. Bhavani and D. Sivaraj, On $\frak{T}$-hyperconnected spaces, Bull. Allahabad Math. Soc., 29 (2014), 15-25.   Google Scholar

[13]

G. Birkhoff, Lattice Theory, Amer. Math. Soc. Colloq. Publ. 25, Providence, RI, 1967. Google Scholar

[14] T. S. Blyth and M. F. Janowitz, Residuation Theory, Pergamon Press, Oxford, 1972.   Google Scholar
[15]

M. Bognár, On some variants of connectedness, Acta Math. Hungar., 79 (1998), 117-122.  doi: 10.1023/A:1006561806310.  Google Scholar

[16]

C. R. Borges, Hyperconnectivity of hyperspaces, Math. Japon., 30 (1985), 757-761.   Google Scholar

[17]

M. K. Bose and R. Tiwari, $(\omega)$topological connectedness and hyperconnectedness, Note Mat., 31 (2011), 93-101.   Google Scholar

[18]

N. Bourbaki, General Topology, Chap 1–4, Springer-Verlag, Berlin, 1989.  Google Scholar

[19]

N. Bourbaki, Éléments de Mathématique, Algébre Commutative, Chap. 1–4, Springer, Berlin, 2006. Google Scholar

[20]

S. Buglyó and Á. Száz, A more important Galois connection between distance functions and inequality relations, Sci. Ser. A Math. Sci. (N.S.), 18 (2009), 17-38.   Google Scholar

[21]

G. Cantor, Über unedliche, linearen Punktmannigfaltigkeiten, Math. Ann., 21 (1983), 545-591.  doi: 10.1007/BF01446819.  Google Scholar

[22]

E. Čech, Topological Spaces, Academia, Prague, 1966. Google Scholar

[23]

C. Chattopadhyay, Debse sets, nowhere dense sets and an ideal in generalized closure spaces, Mat. Vesnik, 59 (2007), 181-188.   Google Scholar

[24]

A. R. Choudhury, A. Mukharjee and M. K. Bose, Hyperconnectedness and extremal disconnectedness in $(\alpha)$topological spaces, Hacet. J. Math. Stat., 44 (2015), 289-294.   Google Scholar

[25]

P. J. Collins, On uiform connection properties, Amer. Math. Monthly, 78 (1971), 372-374.  doi: 10.1080/00029890.1971.11992762.  Google Scholar

[26]

W. W. Comfort and S. Garcia-Ferreira, Resolvability: A selective survey and some new results, Topology Appl., 74 (1996), 149-167.  doi: 10.1016/S0166-8641(96)00052-1.  Google Scholar

[27] Á. Császár, Foundations of General Topology, Pergamon Press, London, 1963.   Google Scholar
[28]

Á. Császár, General Topology, Adam Hilger, Bristol, 1978.  Google Scholar

[29]

Á. Császár, $\gamma$-connected sets, Acta Math. Hungar., 101 (2003), 273-279.  doi: 10.1023/B:AMHU.0000004939.57085.9e.  Google Scholar

[30]

Á. Császár, Extremally disconnected generalized topologies, Ann. Univ. Sci Budapest, 47 (2004), 91-96.   Google Scholar

[31]

Á. Császár and R. Z. Domiaty, Fine quasi-uniformities, Ann. Univ. Budapest, 22/23 (1979/1980), 151-158.   Google Scholar

[32]

Curtis, D. W. and Mathews, J. C., Generalized uniformities for pairs of spaces, Topology Conference, Arizona State University, Tempe, Arizona, 1967,212–246.  Google Scholar

[33] B. A. Davey and H. A. Priestley, Introduction to Lattices and Order, Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511809088.  Google Scholar
[34]

A. S. Davis, Indexed systems of neighbordoods for general topological spaces, Amer. Math. Monthly, 68 (1961), 886-894.  doi: 10.1080/00029890.1961.11989785.  Google Scholar

[35]

J. Deák, A counterexample on completeness in relator spaces, Publ. Math. Debrecen, 41 (1992), 307-309.   Google Scholar

[36]

K. Denecke, M. Erné and S. L. Wismath (Eds.), Galois Connections and Applications, Kluwer Academic Publisher, Dordrecht, 2004. Google Scholar

[37]

D. Doičinov, A unified theory of topological spaces, proximity spaces and uniform spaces, Dokl. Acad. Nauk SSSR, 156 (1964), 21–24. (Russian)  Google Scholar

[38]

J. Dontchev, On superconnected spaces, Serdica, 20 (1994), 345-350.   Google Scholar

[39]

J. Dontchev, On door spaces, Indian J. Pure Appl. Math., 26 (1995), 873-881.   Google Scholar

[40]

J. Dontchev, On submaximal spaces, Tamkang J. Math., 26 (1995), 243-250.   Google Scholar

[41]

J. Dontchev, M. Ganster, G. J. Kennedy and S. D. McCartan, On minimal door, minimal anti-compact and minimal $T_{3/4}$–spaces, Math. Proc. Royal Irish Acad., 98 (1998), 209-215.   Google Scholar

[42]

J. Dontchev, M. Ganster and D. Rose, Ideal resolvability, Topology Appl., 93 (1999), 1-16.  doi: 10.1016/S0166-8641(97)00257-5.  Google Scholar

[43]

E. K. van Douwen, Applications of maximal topologies, Top. Appl., 51 (1993), 125-139.  doi: 10.1016/0166-8641(93)90145-4.  Google Scholar

[44]

L. Dridi, S. Lazaar and T. Turki, $F$-door spaces and $F$-submaximal spaces, Appl. Gen. Topol., 14 (2013), 97-113.  doi: 10.4995/agt.2013.1621.  Google Scholar

[45]

Z. Duszyński, On some concepts of weak connectedness of topological spaces, Acta Math. Hungar., 110 (2006), 81-90.  doi: 10.1007/s10474-006-0008-x.  Google Scholar

[46]

V. A. Efremovič, The geometry of proximity, Mat. Sb., 31 (1952), 189–200. (Russian)  Google Scholar

[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)  Google Scholar

[48]

E. Ekici, Generalized hyperconnectedness, Acta Math. Hungar., 133 (2011), 140-147.  doi: 10.1007/s10474-011-0086-2.  Google Scholar

[49]

E. Ekici, Generalized submaximal spaces, Acta Math. Hungar., 134 (2012), 132-138.  doi: 10.1007/s10474-011-0109-z.  Google Scholar

[50]

E. Ekici and T. Noiri, Connectedness in ideal topological spaces, Novi Sad J. Math., 38 (2008), 65-70.   Google Scholar

[51]

E. Ekici and T. Noiri, *-hyperconnected ideal topological spaces, An. Stiint. Univ. Al. I. Cuza Iasi, 58 (2012), 121-129.   Google Scholar

[52]

R. Engelking, General Topology, Polish Scientific Publishers, Warszawa, 1977.  Google Scholar

[53]

U. V. Fattech and D. Singh, Some results on locally hyperconnected spaces, Ann. Soc. Sci. Bruxelles, Sér. I, 97 (1983), 3-9.   Google Scholar

[54]

U. V. Fattech and D. Singh, A note on $D$-spaces, Bull. Calcutta Math. Soc., 75 (1983), 363–368.  Google Scholar

[55]

P. Fletcher and W. F. Lindgren, Quasi-Uniform Spaces, Marcel Dekker, New York, 1982.  Google Scholar

[56]

G. B. Folland, A tale of topology, Amer. Math. Monthly, 117 (2010), 663-672.  doi: 10.4169/000298910x515730.  Google Scholar

[57] S. A. Gaal, Point Set Topology, Academic Press, New York, 1964.   Google Scholar
[58]

M. Ganster, Preopen sets and resolvable spaces, Kyungpook Math. J., 27 (1987), 135-143.   Google Scholar

[59]

B. Ganter and R. Wille, Formal Concept Analysis, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-642-59830-2.  Google Scholar

[60]

M. Ganster, I. L. Reilly and M. K. Vamanamurthy, Dense sets and irresolvable spaces, Ricerche Mat., 36 (1987), 163-170.   Google Scholar

[61]

B. Garai and C. Bandyopadhyay, Nowhere dense sets and hyperconnected $s$-topological spaces, Bull. Cal. Math. Soc., 92 (2000), 55-58.   Google Scholar

[62]

B. Garai and C. Bandyopadhyay, On parirwise hyperconnected spaces, Soochow J. Math., 27 (2001), 391-399.   Google Scholar

[63]

B. Garai and C. Bandyopadhyay, On irresolvable spaces, Bull. Cal. Math. Soc., 95 (2003), 107-112.   Google Scholar

[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.  Google Scholar

[65]

T. Glavosits, Generated preorders and equivalences, Acta Acad. Paed. Agrienses, Sect. Math., 29 (2002), 95-103.   Google Scholar

[66]

T. Glavosits, Preorders and equivalences generated by commuting relations, Acta Math. Acad. Paedagog. Nyházi. (N.S.), 18 (2002), 53-56.   Google Scholar

[67]

R. G. D. Gnanam, Generalized hyper connected space in bigeneralized topological space, Int. J. Math. Trends Technology, 47 (2017), 27-103.   Google Scholar

[68]

S. W. Golomb, A connected topology for the integers, Amer. Math. Monthly, 66 (1959), 663-665.  doi: 10.1080/00029890.1959.11989385.  Google Scholar

[69]

V. Gregori and J. Ferrer, Quasi-metrization and completion for Pervin's quasi-uniformity, Stohastica, 6 (1982), 151-156.   Google Scholar

[70]

J. A. Guthrie, D. F. Reynolds and and H. E. Stone, Connected expansions of topologies, Bull. Austral. Math. Soc., 9 (1973), 259-265.  doi: 10.1017/S000497270004315X.  Google Scholar

[71]

F. Hausdorff, Grundzüge der Mengenlehre, (German) Chelsea Publishing Company, New York, N. Y., 1949.  Google Scholar

[72]

H. Herrlich, Topological Structeres, Math. Centre Tracts, 52 (1974), 59-122.   Google Scholar

[73]

E. Hewitt, A problem of set-theoretic topology, Duke Math. J, 10 (1943), 309-333.  doi: 10.1215/S0012-7094-43-01029-4.  Google Scholar

[74]

W. Hunsaker and W. Lindgren, Construction of quasi-uniformities, Math. Ann., 188 (1970), 39-42.  doi: 10.1007/BF01435413.  Google Scholar

[75]

A. Illanes, Finite and $\omega$-resolvability, Proc. Amer. Math. Soc., 124 (1996), 1243-1246.  doi: 10.1090/S0002-9939-96-03348-5.  Google Scholar

[76]

J. R. Isbell, Uniform Spaces, Amer. Math. Soc., Providence, 1964. Google Scholar

[77]

S. Jafari and T. Noiri, Properties of $\beta$-connected spaces, Acta Math. Hungar., 101 (2003), 227-236.  doi: 10.1023/B:AMHU.0000003907.90823.79.  Google Scholar

[78]

I. M. James, Topological and Uniform Structures, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-4716-6.  Google Scholar

[79]

A. Kalapodi, Examples on irresolvability, Scientiae Math. Japon., 76 (2013), 461-469.   Google Scholar

[80]

J. L. Kelley, General Topology, Van Nostrand Reinhold Company, New York, 1955.  Google Scholar

[81]

H. Kenyon, Two theorems on relations, Trans. Amer. Math. Soc., 107 (1963), 1-9.  doi: 10.1090/S0002-9947-1963-0148590-2.  Google Scholar

[82]

E. Khalimsky, R. Kopperman and P. R. Meyer, Computer graphics and connected topologies on finite ordered sets, Topology Appl., 36 (1990), 1-17.  doi: 10.1016/0166-8641(90)90031-V.  Google Scholar

[83]

H. J. Kowalsky, Topologische Räumen, Birkhäuser, Basel, 1960. Google Scholar

[84]

M. K. R. S. V. Kumar, Hyperconnected type spaces, Acta Cienc. Indica Math., 31 (2005), 273-275.   Google Scholar

[85] K. Kuratowski, Topologie I, Revised and augmented eddition: Topology I, Academic Press, New York, 1966.   Google Scholar
[86]

J. Kurdics, A note on connection properties, Acta Math. Acad. Paedagog. Nyházi., 12, (1990), 57–59. Google Scholar

[87]

J. Kurdics, Connected and Well-Chained Relator Spaces, Doctoral Dissertation, Lajos Kossuth University, Debrecen, 1991, 30 pp. (Hungarian) Google Scholar

[88]

J. Kurdics, J. Mala and Á. Száz, Connectedness and well-chainedness properties of symmetric covering relators, Pure Math. Appl., 2 (1991), 189-197.   Google Scholar

[89]

J. Kurdics and Á. Száz, Connected relator spaces, Publ. Math. Debrecen, 40 (1992), 155-164.   Google Scholar

[90]

J. Kurdics and Á. Száz, Well-chained relator spaces, Kyungpook Math. J., 32 (1992), 263–271.  Google Scholar

[91]

J. Kurdics and Á. Száz, Well-chainedness characterizations of connected relators, Math. Pannon., 4 (1993), 37-45.   Google Scholar

[92]

R. E. Larson, Minimum and maximum topological spaces, Bull. Acad. Polon., 18 (1970), 707–710.  Google Scholar

[93]

S. W. Lee, M. A. Moon and M. H. Cho, On submaximal and quasi-submaximal spaces, Honam Math. J., 32 (2010), 643-649.  doi: 10.5831/HMJ.2010.32.4.643.  Google Scholar

[94]

J. E. Leuschen and B. T. Sims, Stronger forms of connectivity, Rend. Circ. Mat. Palermo, 21 (1972), 255-266.  doi: 10.1007/BF02843790.  Google Scholar

[95]

N. Levine, Strongly connected sets in topology, Amer. Math. Monthly, 72 (1965), 1098-1101.  doi: 10.2307/2315958.  Google Scholar

[96]

N. Levine, The superset topology, Amer. Math. Monthly, 75 (1968), 745-746.   Google Scholar

[97]

N. Levine, Dense topologies, Amer. Math. Monthly, 75 (1968), 847-852.  doi: 10.1080/00029890.1968.11971077.  Google Scholar

[98]

N. Levine, On uniformities generated by equivalence relations, Rend. Circ. Mat. Palermo, 18 (1969), 62-70.  doi: 10.1007/BF02888946.  Google Scholar

[99]

N. Levine, On Pervin's quasi uniformity, Math. J. Okayama Univ., 14 (1970), 97-102.   Google Scholar

[100]

N. Levine, Well-chained uniformities, Kyungpook Math. J., 11 (1971), 143-149.   Google Scholar

[101]

N. Levine, The finite square semi-uniformity, Kyungpook Math. J., 13 (1973), 179-184.   Google Scholar

[102]

S. N. Maheswari and U. Tapi, Connectedness of a stronger type in topological spaces, Nanta Math., 12 (1979), 102-109.   Google Scholar

[103]

R. A. Mahmoud and D. A. Rose, A note on spaces via dense sets, Tamkang J. Math., 24 (1993), 333-339.   Google Scholar

[104]

R. A. Mahmoud and D. A. Rose, A note on submaximal spaces and SMPC functions, Demonstratio Math., 28 (1995), 567-573.   Google Scholar

[105]

J. Mala, An equation for families of relations, Pure Math. Apll., Ser. B, 1 (1990), 185-188.   Google Scholar

[106]

J. Mala, Relator Spaces, Doctoral Dissertation, Lajos Kossuth University, Debrecen, 1990, 48 pp. (Hungarian) Google Scholar

[107]

J. Mala, Relators generating the same generalized topology, Acta Math. Hungar., 60 (1992), 291-297.  doi: 10.1007/BF00051647.  Google Scholar

[108]

J. Mala, Finitely generated quasi-proximities, Period. Math. Hungar., 35 (1997), 193-197.  doi: 10.1023/A:1004553417107.  Google Scholar

[109]

J. Mala, On proximal properties of proper symmetrizations of relators, Publ. Math. Debrecen, 58 (2001), 1-7.   Google Scholar

[110]

J. Mala and Á. Száz, Equations for families of relations can also be solved, C. R. Math. Rep. Acad. Sci. Canada, 12 (1990), 109-112.   Google Scholar

[111]

J. Mala and Á. Száz, Properly topologically conjugated relators, Pure Math. Appl., Ser. B, 3 (1992), 119-136.   Google Scholar

[112]

J. Mala and Á. Száz, Modifications of relators, Acta Math. Hungar., 77 (1997), 69-81.  doi: 10.1023/A:1006583622770.  Google Scholar

[113]

Z. P. Mamuzić, Introduction to General Topology, Noordhoff, Groningen, 1963.  Google Scholar

[114]

J. C. Mathews, A note on well-chained spaces, Amer. Math. Monthly, 75 (1968), 273-275.  doi: 10.2307/2314959.  Google Scholar

[115]

P. M. Mathew, On hyperconnected spaces, Indian J. Pure Appl. Math., 19 (1988), 1180–1184.  Google Scholar

[116]

P. M. Mathew, On ultraconnected spaces, Int. J. Math. Sci., 13 (1990), 349-352.  doi: 10.1155/S0161171290000515.  Google Scholar

[117]

J. C. Mathews, A note on well-chained spaces, Amer. Math. Monthly, 75 (1968), 273-275.  doi: 10.2307/2314959.  Google Scholar

[118]

S. D. McCartan, Door spaces are identifiable, Proc. Roy. Irish Acad., 87A (1987), 13-16.   Google Scholar

[119]

S. Modak, Relativization in resolvability and irresolvability, Int. Math. Forum, 6 (2011), 1059-1064.   Google Scholar

[120]

S. G. Mrówka and W. J. Pervin, On uniform connectedness, Proc. Amer. Mth. Soc., 15 (1964), 446-449.  doi: 10.1090/S0002-9939-1964-0161307-7.  Google Scholar

[121]

Á. Münnich and Á. Száz, An alternative theorem for continuous relations and its applications, Publ. Inst. Math. (Beograd), 33 (1983), 163-168.   Google Scholar

[122]

M. G. Murdeshwar and S. A. Naimpally, Quasi-Uniform Topological Spaces, Noordhoff, Groningen, 1966.  Google Scholar

[123]

L. Nachbin, Topology and Order, D. Van Nostrand, Princeton, 1965.  Google Scholar

[124] S. A. Naimpally and B. D. Warrack, Proximity Spaces, Cambridge University Press, Cambridge, 1970.   Google Scholar
[125]

H. Nakano and K. Nakano, Connector theory, Pacific J. Math., 56 (1975), 195-213.  doi: 10.2140/pjm.1975.56.195.  Google Scholar

[126]

G. Navalagi, M. L. Thivagar, R. R. Rajeswari and S. A. Ponmani, On $(1, 2)\alpha$-hyperconnected spaces, Int. J. Math. Anal., 3 (2006), 121-129.   Google Scholar

[127]

T. Nieminen, On ultrapseudocompact and related spaces, Ann. Acad. Sci. Fennicae, 3 (1977), 185-205.  doi: 10.5186/aasfm.1977.0321.  Google Scholar

[128]

T. Noiri, A note on hyperconnected sets, Mat. Vesnik, 3 (1979), 53-60.   Google Scholar

[129]

T. Noiri, Functions which preserve hyperconnected spaces, Rev. Roum. Math. Pures Appl., 25 (1980), 1091-1094.   Google Scholar

[130]

T. Noiri, Hyperconnectedness and preopen sets, Rev. Roum. Math. Pures Appl., 29 (1984), 329-334.   Google Scholar

[131]

T. Noiri, Properties of hyperconnected spaces, Acta Math. Hungar., 66 (1995), 147-154.  doi: 10.1007/BF01874359.  Google Scholar

[132]

K. Padmavally, An example of a connected irresolvable Hausdorff space, Duke Math. J., 20 (1953), 513-520.  doi: 10.1215/S0012-7094-53-02050-X.  Google Scholar

[133]

W. Page, Topological Uniform Structures, John Wiley and Sons Inc, New York, 1978.  Google Scholar

[134]

G. Pataki, Supplementary notes to the theory of simple relators, Radovi Mat., 9 (1999), 101–118.  Google Scholar

[135]

G. Pataki, On the extensions, refinements and modifications of relators, Math. Balk., 15 (2001), 155-186.   Google Scholar

[136]

G. Pataki, Well-chained, Connected and Simple Relators, Ph.D Dissertation, Debrecen, 2004. Google Scholar

[137]

G. Pataki and Á. Száz, A unified treatment of well-chainedness and connectedness properties, Acta Math. Acad. Paedagog. Nyházi. (N.S.), 19 (2003), 101-165.   Google Scholar

[138]

W. J. Pervin, Uniformizations of neighborhood axioms, Math. Ann., 147 (1962), 313-315.  doi: 10.1007/BF01440952.  Google Scholar

[139]

W. J. Pervin, Quasi-uniformization of topological spaces, Math. Ann., 147 (1962), 316-317.  doi: 10.1007/BF01440953.  Google Scholar

[140]

W. J. Pervin, Quasi-proximities for topological spaces, Math. Ann., 150 (1963), 325-326.  doi: 10.1007/BF01470761.  Google Scholar

[141]

W. J. Pervin, Connectedness in bitopological spaces, Indag. Math., 29 (1967), 369-372.   Google Scholar

[142]

V. Pipitone and G. Russo, Spazi semiconnessi e spazi semiaperty, Rend. Circ. Mat. Palermo, 24 (1975), 273-285.  doi: 10.1007/BF02843735.  Google Scholar

[143]

Cs. Rakaczki and Á. Száz, Semicontinuity and closedness properties of relations in relator spaces, Mathematica (Cluj), 45 (2003), 73-92.   Google Scholar

[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. Google Scholar

[145]

V. Renukadevi, On generalizations of hyperconnected spaces, J. Adv. Res. Pure Math., 4 (2012), 46-58.   Google Scholar

[146]

V. Renukadevi, Remarks on generalized hypperconnectedness, Acta Math. Hung., 136 (2012), 157-164.  doi: 10.1007/s10474-011-0192-1.  Google Scholar

[147]

F. Riesz, Die Genesis der Raumbegriffs, Math. Naturwiss. Ber. Ungarn, 24 (1907), 309-353.   Google Scholar

[148]

F. Riesz, Stetigkeitsbegriff und abstrakte Mengenlehre, Atti Ⅳ Congr. Intern.Mat., Roma, Ⅱ (1908), 18-24.   Google Scholar

[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.  Google Scholar

[150]

H. M. Salih, On door spaces, Journal of College of Education, Al-Mustansnyah University, Bagdad, Iraq, 3 (2006), 112–117. Google Scholar

[151]

J. Schröder, On sub-, pseeudo- and quasimaximalspaces, Comm. Math. Univ. Carolinae, 39 (1998), 197-206.   Google Scholar

[152]

J. Schröder, Some answers concerning submaximal spaces, Questions and Answers in General Topology, 17 (1999), 221-225.   Google Scholar

[153]

A. K. Sharma, On some properties of hyperconnected spaces, Mat. Vesnik, 14 (1977), 25-27.   Google Scholar

[154]

R.-X. Shen, A note on generalized connectedness, Acta Math. Hungar., 122 (2009), 231-235.  doi: 10.1007/s10474-008-8009-6.  Google Scholar

[155]

J. L. Sieber and W. J. Pervin, Connectedness in syntopogeneous spaces, Proc. Amer. Math. Soc., 15 (1964), 590-595.  doi: 10.1090/S0002-9939-1964-0166747-8.  Google Scholar

[156]

W. Sierpinski, General Topology, Mathematical Expositions, 7, University of Toronto Press, Toronto, 1952.  Google Scholar

[157]

Yu. M. Smirnov, On proximity spaces, Math. Sb., 31 (1952), 543–574. (Russian.)  Google Scholar

[158]

L. A. Steen and J. A. Seebach, Counterexamples in Topology, Springer-Verlag, New York, 1970.  Google Scholar

[159]

A. K. Steiner, The lattice of topologies: Structure and complementation, Trans. Amer. Math. Soc., 122 (1966), 379-398.  doi: 10.1090/S0002-9947-1966-0190893-2.  Google Scholar

[160]

Gy. Szabó and Á. Száz, Defining nets for integration, Publ. Math. Debrecen, 36 (1989), 237-252.   Google Scholar

[161]

Á. Száz, Coherences instead of convergences, Proc. Conf. Convergence and Generalized Functions (Katowice, Poland, 1983), Polish Acad Sci., Warsaw, 1984,141–148. Google Scholar

[162]

Á. Száz, Basic tools and mild continuities in relator spaces, Acta Math. Hungar., 50 (1987), 177-201.  doi: 10.1007/BF01903935.  Google Scholar

[163]

Á. Száz, Directed, topological and transitive relators, Publ. Math. Debrecen, 35 (1988), 179–196.  Google Scholar

[164]

Á. Száz, Projective and inductive generations of relator spaces, Acta Math. Hungar., 53 (1989), 407-430.  doi: 10.1007/BF01953378.  Google Scholar

[165]

Á. Száz, Lebesgue relators, Monatsh. Math., 110 (1990), 315-319.  doi: 10.1007/BF01301684.  Google Scholar

[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. Google Scholar

[167]

Á. Száz, Relators, Nets and Integrals, Unfinished doctoral thesis, Debrecen, 1991. Google Scholar

[168]

Á. Száz, Inverse and symmetric relators, Acta Math. Hungar., 60 (1992), 157-176.  doi: 10.1007/BF00051766.  Google Scholar

[169]

Á. Száz, Structures derivable from relators, Singularité, 3 (1992), 14-30.   Google Scholar

[170]

Á. Száz, Refinements of relators, Tech. Rep., Inst. Math., Univ. Debrecen, 76 (1993), 19 pp. Google Scholar

[171]

Á. Száz, Cauchy nets and completeness in relator spaces, Colloq. Math. Soc. János Bolyai, 55 (1993), 479-489.   Google Scholar

[172]

Á. Száz, Neighbourhood relators, Bolyai Soc. Math. Stud., 4 (1995), 449-465.   Google Scholar

[173]

Á. Száz, Relations refining and dividing each other, Pure Math. Appl. Ser. B, 6 (1995), 385-394.   Google Scholar

[174]

Á. Száz, Connectednesses of refined relators, Tech. Rep., Inst. Math., Univ. Debrecen, 1996, 6 pp. Google Scholar

[175]

Á. Száz, Topological characterizations of relational properties, Grazer Math. Ber., 327 (1996), 37-52.   Google Scholar

[176]

Á. Száz, Uniformly, proximally and topologically compact relators, Math. Pannon., 8 (1997), 103-116.   Google Scholar

[177]

Á. Száz, An extension of Kelley's closed relation theorem to relator spaces, Filomat, 14 (2000), 49-71.   Google Scholar

[178]

Á. Száz, Somewhat continuity in a unified framework for continuities of relations, Tatra Mt. Math. Publ., 24 (2002), 41-56.   Google Scholar

[179]

Á. Száz, A Galois connection between distance functions and inequality relations, Math. Bohem., 127 (2002), 437-448.   Google Scholar

[180]

Á. Száz, Upper and lower bounds in relator spaces, Serdica Math. J., 29 (2003), 239-270.   Google Scholar

[181]

Á. Száz, An extension of Baire's category theorem to relator spaces, Math. Morav., 7 (2003), 73-89.  doi: 10.5937/matmor0307073s.  Google Scholar

[182]

Á. Száz, Rare and meager sets in relator spaces, Tatra Mt. Math. Publ., 28 (2004), 75-95.   Google Scholar

[183]

Á. Száz, Galois-type connections on power sets and their applications to relators, Tech. Rep., Inst. Math., Univ. Debrecen, 2 (2005), 38 pp. Google Scholar

[184]

Á. Száz, Supremum properties of Galois–type connections, Comment. Math. Univ. Carolin., 47 (2006), 569-583.   Google Scholar

[185]

Á. Száz, Minimal structures, generalized topologies, and ascending systems should not be studied without generalized uniformities, Filomat, 21 (2007), 87-97.  doi: 10.2298/FIL0701087S.  Google Scholar

[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. Google Scholar

[187]

Á. Száz, Galois type connections and closure operations on preordered sets, Acta Math. Univ. Comen., 78 (2009), 1-21.   Google Scholar

[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.  Google Scholar

[189]

Á. Száz, Foundations of the theory of vector relators, Adv. Stud. Contemp. Math., 20 (2010), 139-195.   Google Scholar

[190]

Á. Száz, Galois-type connections and continuities of pairs of relations, J. Int. Math. Virt. Inst., 2 (2012), 39-66.   Google Scholar

[191]

Á. Száz, Lower semicontinuity properties of relations in relator spaces, Adv. Stud. Contemp. Math., (Kyungshang), 23 (2013), 107-158.   Google Scholar

[192]

Á. Száz, An extension of an additive selection theorem of Z. Gajda and R. Ger to vector relator spaces, Sci. Ser. A Math. Sci. (N.S.), 24 (2013), 33-54.   Google Scholar

[193]

Á. Száz, Inclusions for compositions and box products of relations, J. Int. Math. Virt. Inst., 3 (2013), 97-125.   Google Scholar

[194]

Á. Száz, A particular Galois connection between relations and set functions, Acta Univ. Sapientiae, Math., 6 (2014), 73-91.  doi: 10.2478/ausm-2014-0019.  Google Scholar

[195]

Á. Száz, Generalizations of Galois and Pataki connections to relator spaces, J. Int. Math. Virt. Inst., 4 (2014), 43-75.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[198]

Á. Száz, A natural Galois connection between generalized norms and metrics, Acta Univ. Sapientiae Math., 9 (2017), 360-373.  doi: 10.1515/ausm-2017-0027.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[201]

Á. Száz, Contra continuity properties of relations in relator spaces, Tech. Rep., Inst. Math., Univ. Debrecen, 5 (2017), 48 pp. Google Scholar

[202]

Á. Száz, The closure-interior Galois connection and its applications to relational equations and inclusions, J. Int. Math. Virt. Inst., 8 (2018), 181-224.   Google Scholar

[203]

Á. Száz, A unifying framework for studying continuity, increasingness, and Galois connections, MathLab J., 1 (2018), 154-173.   Google Scholar

[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.  Google Scholar

[205]

Á. Száz, Relationships between inclusions for relations and inequalities for corelations, Math. Pannon., 26 (2018), 15-31.   Google Scholar

[206]

Á. Száz, Galois and Pataki connections on generalized ordered sets, Earthline J. Math. Sci., 2 (2019), 283-323.   Google Scholar

[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. Google Scholar

[208]

Á. Száz and J. Túri, Comparisons and compositions of Galois–type connections, Miskolc Math. Notes, 7 (2006), 189-203.  doi: 10.18514/MMN.2006.144.  Google Scholar

[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.  Google Scholar

[210]

J. P. Thomas, Maximal connected topologies, J. Austral. Math. Soc., 8 (1968), 700-705.  doi: 10.1017/S1446788700006510.  Google Scholar

[211]

T. Thompson, Characterizations of irreducible spaces, Kyungpook Math. J., 21 (1981), 191–194.  Google Scholar

[212]

W. J. Thron, Topological Structures, Holt, Rinehart and Winston, New York, 1966.  Google Scholar

[213]

H. Tietze, Beiträge zur allgemeinen Topologie I. Axiome für verschiedene Fassungen des Umgebungsbegriffs, Math. Ann., 88 (1923), 290-312.  doi: 10.1007/BF01579182.  Google Scholar

[214] J. W. Tukey, Convergence and Uniformity in Topology, Princeton Univ. Press, Princeton, 1940.   Google Scholar
[215]

A. Weil, Sur les Espaces á Structure Uniforme et sur la Topologie Générale, Actual. Sci. Ind. 551, Herman and Cie, Paris, 1937. Google Scholar

[216]

R. L. Wilder, Evolution of the topological concept of connected, Amer. Math. Monthly, 85 (1978), 720-726.  doi: 10.2307/2321676.  Google Scholar

[217]

J. Wu, C. Wang and D. Zhang, Connected door spaces and topological solutions of equations, Aequationes Math., 92 (2018), 1149-1161.  doi: 10.1007/s00010-018-0577-0.  Google Scholar

[218]

I. Zorlutuna, On a function of hyperconnected spaces, Demonstr. Math., 40 (2007), 939-950.   Google Scholar

[1]

G. Conner, Christopher P. Grant, Mark H. Meilstrup. A Sharkovsky theorem for non-locally connected spaces. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3485-3499. doi: 10.3934/dcds.2012.32.3485
[2]

Luiz Gustavo Farah. Local solutions in Sobolev spaces and unconditional well-posedness for the generalized Boussinesq equation. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1521-1539. doi: 10.3934/cpaa.2009.8.1521
[3]

Zhichun Zhai. Well-posedness for two types of generalized Keller-Segel system of chemotaxis in critical Besov spaces. Communications on Pure & Applied Analysis, 2011, 10 (1) : 287-308. doi: 10.3934/cpaa.2011.10.287
[4]

Richard Miles, Thomas Ward. Directional uniformities, periodic points, and entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3525-3545. doi: 10.3934/dcdsb.2015.20.3525
[5]

Piotr Gwiazda, Agnieszka Świerczewska-Gwiazda, Aneta Wróblewska. Generalized Stokes system in Orlicz spaces. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2125-2146. doi: 10.3934/dcds.2012.32.2125
[6]

Markus Banagl. Singular spaces and generalized Poincaré complexes. Electronic Research Announcements, 2009, 16: 63-73. doi: 10.3934/era.2009.16.63
[7]

Tao Chen, Linda Keen. Slices of parameter spaces of generalized Nevanlinna functions. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5659-5681. doi: 10.3934/dcds.2019248
[8]

Ji Gao. On the generalized pythagorean parameters and the applications in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 557-567. doi: 10.3934/dcdsb.2007.8.557
[9]

Tadahiro Oh, Yuzhao Wang. On global well-posedness of the modified KdV equation in modulation spaces. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2971-2992. doi: 10.3934/dcds.2020393
[10]

Qunyi Bie, Qiru Wang, Zheng-An Yao. On the well-posedness of the inviscid Boussinesq equations in the Besov-Morrey spaces. Kinetic & Related Models, 2015, 8 (3) : 395-411. doi: 10.3934/krm.2015.8.395
[11]

Zhaohui Huo, Boling Guo. The well-posedness of Cauchy problem for the generalized nonlinear dispersive equation. Discrete & Continuous Dynamical Systems, 2005, 12 (3) : 387-402. doi: 10.3934/dcds.2005.12.387
[12]

Yongye Zhao, Yongsheng Li, Wei Yan. Local Well-posedness and Persistence Property for the Generalized Novikov Equation. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 803-820. doi: 10.3934/dcds.2014.34.803
[13]

Aneta Wróblewska-Kamińska. Unsteady flows of non-Newtonian fluids in generalized Orlicz spaces. Discrete & Continuous Dynamical Systems, 2013, 33 (6) : 2565-2592. doi: 10.3934/dcds.2013.33.2565
[14]

Ren-You Zhong, Nan-Jing Huang. Strict feasibility for generalized mixed variational inequality in reflexive Banach spaces. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 261-274. doi: 10.3934/naco.2011.1.261
[15]

Anna Anop, Robert Denk, Aleksandr Murach. Elliptic problems with rough boundary data in generalized Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (2) : 697-735. doi: 10.3934/cpaa.2020286
[16]

Shouming Zhou. The Cauchy problem for a generalized $b$-equation with higher-order nonlinearities in critical Besov spaces and weighted $L^p$ spaces. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4967-4986. doi: 10.3934/dcds.2014.34.4967
[17]

Sergey Zelik, Jon Pennant. Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$. Communications on Pure & Applied Analysis, 2013, 12 (1) : 461-480. doi: 10.3934/cpaa.2013.12.461
[18]

Hartmut Pecher. Almost optimal local well-posedness for the Maxwell-Klein-Gordon system with data in Fourier-Lebesgue spaces. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3303-3321. doi: 10.3934/cpaa.2020146
[19]

Nikolaos Bournaveas. Local well-posedness for a nonlinear dirac equation in spaces of almost critical dimension. Discrete & Continuous Dynamical Systems, 2008, 20 (3) : 605-616. doi: 10.3934/dcds.2008.20.605
[20]

Fucai Li, Yanmin Mu, Dehua Wang. Local well-posedness and low Mach number limit of the compressible magnetohydrodynamic equations in critical spaces. Kinetic & Related Models, 2017, 10 (3) : 741-784. doi: 10.3934/krm.2017030

 Impact Factor: 0.263

Tools

Metrics

  • PDF downloads (241)
  • HTML views (567)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences