Celebrating Professor Themba A. Dube (A TAD Celebration I)

Document Type : Research Paper

Author

Department of Mathematical Sciences, University of South Africa, P.O. Box 392, Tshwane, UNISA 0003, South Africa.\\ National Institute for Theoretical and Computational Sciences (NITheCS), Johannesburg, South Africa.

10.48308/cgasa.2023.234071.1453

Abstract

This is the first in a series of survey papers featuring the mathematical contributions of Themba Dube to pointfree topology and ordered algebraic structures. We cover Dube’s distinguished career and benefactions to the discipline with the early beginnings in nearness frames. We envelope the essential aspects of Dube’s work in structured frames. The paper radars across the initial themes of nearness, metrization, and uniform structures that Dube conceives and presents in his independent and joint published papers. Pertinent subcategories of these structured frames are discussed. We also feature Dube’s imprints on certain categorical aspects of his work on βL, λL, υL and ßL.

Keywords

Main Subjects


[1] Adjei, I. and Dube, T., The Banaschewski extension and some variants of openness, Houston Journal of Mathematics 47, no.1, 2021, 245 - 261.
[2] Banaschewski, B., Dube, T., Gilmour, C. and Walters-Wayland, J., Oz in pointfree topology, Quaestiones Mathematicae, 32(2) (2009), 215 - 227.
[3] Bayih, T., Dube, T. and Ighedo, O., On the Menger and almost Menger properties in locales, Applied General Topology 22, no. 1 (2021), 199 - 221.
[4] Bhattacharjee, P. and Dube, T., On fraction-dense algebraic frames, Algebra Universalis (2022), 83:6,18 pages.
[5] Bhattacharjee, P. and Dube, T., On the sobriety of the inverse topology, Algebra Universalis 76 (2016), 445-454.
[6] Dube, T., Structures in Frames, PhD thesis, University of Durban-Westville (1992).
[7] Dube, T., Separability in locales, Quaestiones Mathematicae 17 (1994), no. 3, 333-338.
[8] Dube, T., The Tamano-Dowker type theorems for nearness frames, Journal of Pure and Applied Algebra 99 (1995) l - 7.
[9] Dube, T., Paracompact and locally fine nearness frames, Topology and its Applications 62 (1995), no. 3, 247–253.
[10] Dube, T., A short note on separable frames, Commentationes Mathematicae Universitatis Carolinae 37,2 (1996) 375-377.
[11] Dube, T., A note on complete regularity and normality, Quaestiones Mathematicae 19 (1996), no. 3-4, 467-478.
[12] Dube, T., Strong nearness frames, Proceedings Symposium on Categorical Topology (Rondebosch, 1994), 103–112, Univ. Cape Town, Rondebosch, 1999.
[13] Dube, T., Sigma-compactness via Nearness, Kyungpook Mathematical Journal. (39) 1999, 207 - 214.
[14] Dube, T. and Valov, V., Generalized tri-quotient maps and Cech-completeness, Commentationes Mathematicae Universitatis Carolinae, 42,1 (2001) 187-194.
[15] Dube, T. Balanced and closed-generated filters in frames, Quaestiones Mathematicae 26 (2003), 73 - 81.
[16] Dube, T., On Compactness of Frames, Algebra Universalis 51 (2004) 411 – 417.
[17] Dube, T., Irreducibility in pointfree topology, Quaestiones Mathematicae 27 (2004), no. 3, 231-241.
[18] Dube, T., Notes on uniform frames, Quaestiones Mathematicae 27 (2004), 9 - 20.
[19] Dube, T., Bounded quotients of frames, Quaestiones Mathematicae 28 (2005), 55-72.
[20] Dube, T., Submaximality in locales, Topology Proceedings 29 No. 2 (2005), pp. 431-444.
[21] Dube, T., An algebraic view of weaker forms of realcompactness, Algebra Universalis 55 (2006) 187 - 202.
[22] Dube, T., Katetov revisited: a frame-theoretic excursion, Quaestiones Mathematicae 30 (2007), 365 – 380.
[23] Dube, T., Pointfree functional compactness, Acta Mathematica Hungarica, 116(3) (2007), 223 - 237.
[24] Dube, T. and Matutu, P., Pointfree pseudocompactness revisited, Topology and its Applications 154 (2007) 2056-2062.
[25] Dube, T. and Matutu, P., A few points on pointfree pseudocompactness, Quaestiones Mathematicae 30 (2007), no. 4, 451 - 464.
[26] Dube, T. and Walters-Wayland, J., Coz-onto Frame Maps and Some Applications, Applied Categorical Structures (2007) 15: 119 - 133.
[27] Dube, T. and Walters-Wayland, J., Weakly Pseudocompact Frames, Applied Categorical Structures (2008) 16: 749-761.
[28] Dube, T., Realcompactness and certain types of subframes, Algebra Universalis 58 (2008) 181 - 202.
[29] Dube, T., A Little More on Coz-Unique Frames, Applied Categorical Structures (2009) 17: 63 - 73.
[30] Dube, T., Remote points and the like in pointfree topology, Acta Mathematica Hungarica 123 (2009), no. 3, 203 - 222.
[31] Dube, T., Some ring-theoretic properties of almost P-frames, Algebra Universalis 60 (2009) 145-162.
[32] Dube, T. and Matlabyana, M., Notes concerning characterizations of quasi-F frames, Quaestiones Mathematicae, 32 (2009), no. 4, 551- 567.
[33] Dube, T., Some algebraic characterizations of F-frames, Algebra Universalis, 62 (2009), 273-28.
[34] Dube, T., Concerning P-frames, essential P-frames, and strongly zero-dimensional frames, Algebra Universalis 61 (2009), no. 1,115 - 138.
[35] Dube, T. and Mugochi, M.M., Zero-dimensionality in structured frames, Far East Journal of Mathematical Sciences 40 (2010), no. 1, 121 - 136.
[36] Dube, T. and Naidoo, I., On openness and surjectivity of lifted frame homomorphisms, Topology and its Applications, 157 (2010), 2159 - 2171.
[37] Dube, T., Notes on Pointfree Disconnectivity with a Ring-theoretic Slant, Applied Categorical Structures 18 (2010), no.1, 55 - 72.
[38] Dube, T., Contracting the socle in rings of continuous functions, Rendiconti del Seminario Matematico della Universit`a di Padova, 123 (2010), 37 - 53.
[39] Dube, T., On the ideal of functions with compact support in pointfree function rings, Acta Mathematica Hungarica, 129(2010), 205 - 226.
[40] Dube, T. and Mugochi, M.M., Thoughts on quotient-fine nearness frames, Applied Categorical Structures, 19 (2011), 511 - 521.
[41] Dube, T. and Mugochi, M.M., A note on almost uniform nearness frames, Quaestiones Mathematicae, 34(2) (2011), 247 - 263.
[42] Dube, T. and Naidoo, I., Erratum to “On openness and surjectivity of lifted frame homomorphisms”, Topology and its Applications, 157 (2011), 2257 - 2259.
[43] Dube, T. and Matlabyana, M., Concerning variants of C-embedding in pointfree topology, Topology and its Applications, 158 (2011), 2307 - 2321.
[44] Dube, T., A broader view of the almost Lindelof property, Algebra Universalis, 65 (2011), 263 - 276.
[45] Dube, T., Real ideals in pointfree rings of continuous functions, Bulletin of the Australian Mathematical Society, 83 (2011), 338 - 352.
[46] Dube, T., Notes on the socle of certain types of f-rings, Bulletin of the Iranian Mathematical Society, Vol 38, No.2, (2012), 517 - 528.
[47] Dube, T. and Naidoo, I., When lifted frame homomorphisms are closed, Topology and its Applications, 159 (2012), 3049 - 3058.
[48] Dube, T., Extending and contracting maximal ideals in the function rings of pointfree topology, Bulletin Mathematique de la Societe des Sciences Mathematiques de Roumanie Tome 55 (103), No. 4, 2012, 365 - 374.
[49] Dube, T. and Naidoo, I., Round squares in the category of Frames, Houston Journal of Mathematics, 39 (2) (2013), 453-473.
[50] Dube, T., Coherence classes of ideals on normal lattices with applications to C(X), Mathematica Slovaca (2013), Vo. 63, Issue 4, 679 - 692.
[51] Dube, T., A note on relative pseudocompactness in the category of frames, Bulletin of the Australian Mathematical Society (2013), Vol. 87, Issue 01, 120 - 130.
[52] Dube, T., The hull-kernel and inverse topologies as frames, Algebra Universalis (2013) Vol. 70, Issue 2, 197 - 212.
[53] Dube, T., Iliadis, S., van Mill, J. and Naidoo, I., Universal frames, Topology and its Applications 160 (2013), 2454 - 2464.
[54] Dube, T. and Matlabyana, M., Cozero complemented frames, Topology and its Applications, 16 (2013), 1345 - 1352.
[55] Dube, T. and Ighedo, O., Comments regarding d-ideals of certain f-rings, Journal of Algebra and its Applications 12 (2013), 1350008 (16 pages).
[56] Dube, T., Mugochi, M.M. and Naidoo, I., Cech-completeness in pointfree topology, Quaestiones Mathematicae 37:1 (2014), 49 - 65.
[57] Dube, T., Naidoo, I. and Ncube, C., Isocompactness in the category of locales, Applied Categorical Structures 22 (2014), 727 -739.
[58] Dube, T., Iliadis, S., van Mill, J. and Naidoo, I., A pseudocompact completely regular frame which is not spatial, Order (2014) 31: 115 - 120.
[59] Dube, T., Naidoo, I. and Ncube, C., Nearly realcompact frames, Topology and its Applications 168 (2014), 25 - 39.
[60] Dube, T., Naidoo, I. and Ncube, C., On a generalization of pointfree realcompactness, Topology and its Applications 163 (2014), 80 - 92.
[61] Dube, T. and Ighedo, O., Two functors induced by certain ideals of function rings, Applied Categorical Structures 22 (2014), 663 - 681.
[62] Dube, T. and Ighedo, O., On z-ideals of pointfree function rings, Bulletin of the Iranian Mathematical Society 40 (2014), 657 - 675.
[63] Dube, T., Concerning maximal l-ideals of rings of continuous integer-valued functions, Algebra Universalis, 72 (2014), 359 - 370.
[64] Dube, T., Pseudocompact supports in pointfree topology, Houston Journal of Mathematics 40 (2014), 601 - 620.
[65] Dube, T. and Mugochi, M.M., Localic remote points revisited, Filomat, 29(1) (2015), 111 - 120.
[66] Dube, T. and Naidoo, I., More on uniform paracompactness in pointfree topology, Mathematica Slovaca, 65 (2015), 273-288.
[67] Dube, T. and Nsonde Nsayi, J., When rings of continuous functions are weakly regular, Bulletin of the Belgian Mathematical Society, Simon Stevin, 22 (2015), 213-226.
[68] Dube, T. and Nsonde Nsayi, J., When certain prime ideals in rings of continuous functions are minimal or maximal, Topology and its Applications 192 (2015), 98-112.
[69] Dube, T. and Ighedo, O., More ring-theoretic characterizations of P-frames, Journal of Algebra and its Applications 14(5) (2015), 150061 (8 pages).
[70] Dube, T. and Ighedo, O., Covering maximal ideals with minimal primes, Algebra Universalis, 74 (2015), 411 - 424.
[71] Dube, T., Georgiou, D.N., Megaritis, A.C. and Moshokoa, S.P., A study of covering dimension for the class of finite lattices, Discrete Mathematics 338 (2015), 1096-1110.
[72] Dube, T., Ideals associated with realcompactness in pointfree function rings, Quaestiones Mathematicae 38(6) (2015), 885 - 899.
[73] Dube, T. and Nsonde Nsayi, J., A note on spaces that are finitely an F-space, Topology and its Applications 202 (2016), 365-356.
[74] Dube, T. and Ighedo, O., Higher order z-ideals in commutative rings, Miskolc Mathematical Notes 17 (2016), 171-185.
[75] Dube, T. and Ighedo, O., More on locales in which every open sublocale is zembedded, Topology and its Applications 201 (2016), 110-123.
[76] Dube, T. and Ighedo, O., Characterising points which make P-frames, Topology and its Applications 200 (2016), 146-159.
[77] Dube, T. and Nsonde Nsayi, J., Another ring-theoretic characterization of boundary spaces, Houston Journal of Mathematics 42 (2016), 709 - 722.
[78] Dube, T., A note on lattices of z-ideals of f-rings, New York Journal of Mathematics 22 (2016), 351-361.
[79] Dube, T., On maps between Stone- ˇ Cech compactifications induced by lattice homomorphisms, Filomat 30 (2016) 2465-2474.
[80] Dube, T., Georgiou, D.N., Megaritis, A.C. and Sereti, F., Studying the Krull dimension of finite lattices under the prism of matrices, Filomat 31 (2017), 2901-2915.
[81] Dube, T., Naidoo, I. and Nasirzadeh, N., Pseudocompleteness in the category of locales, Topology and its Applications 231 (2017), 113 -127.
[82] Dube, T., A note on weakly pseudocompact locales, Applied General Topology 18 (2017), 131-141.
[83] Dube, T., When spectra of lattices of z-ideals are Stone-Cech compactifications, Mathematica Bohemica 142 (2017), 323 -336.
[84] Dube, T., Commutative rings in which zero-components of essential primes are essential, Journal of Algebra and its Applications 16 (2017), 17502024 (15 pages).
[85] Dube, T., When Boole commutes with Hewitt and Lindel¨of, Applied Categorical Structures 25 (2017), 1097- 1111.
[86] Dube, T. and Ighedo, O., On lattices of z-ideals of function rings, Mathematica Slovaca 68 (2018), 271 -284.
[87] Dube, T. and Ighedo, O., Concerning the summand intersection property in function rings, Houston Journal of Mathematics 44 (2018), 1029 - 1049.
[88] Dube, T., On quasi-normality of function rings, Rocky Mountain Journal of Mathematics 40 (2018) 157-179.
[89] Dube, T., Some connections between frames of radical ideals and frames of z-ideals, Algebra Universalis 79 (2018), 18 pages.
[90] Dube, T., Maximal Lindel¨of locales, Applied Categorical Structures (2019) 27:687–702.
[91] Dube, T., Rings in which sums of d-ideals are d-ideals, Journal of the Korean Mathematical Society 56 (2019), 539-558.
[92] Dube, T., and Sithole, L., On the sublocale of an algebraic frame induced by the d-nucleus, Topology and its Applications 263 (2019), 90 - 106.
[93] Dube, T., On the socle of an algebraic frame, Bulletin Mathematique de la Societe des Sciences Mathematiques de Roumanie Tome 110 (2019), 371 - 385.
[94] Dube, T., First steps going down on algebraic frames, Hacettepe Journal of Mathematical Statistics 48 (2019), 1792-1807.
[95] Dube, T., Concerning P-sublocales and disconnectivity, Applied Categorical Structures 27 (2019), 365 - 383.
[96] Dube, T., Ghirati, M. and Nazari, A., Rings in which idempotents generate maximal or minimal ideals, Algebra Universalis, 81 (2020), article 30.
[97] Dube, T., On the maximal regular ideal of pointfree function rings, and more, Topology and its Applications 273 (2020), 106960.
[98] Dube, T. and Sarpoushi Robat M., On densely normal sublocales, Topology and its Applications 275 (2020), 107015.
[99] Dube, T., Estaji, A.A. and Sarpoushi, Robat M., Some relative normality properties in locales, Mathematica Slovaca, 70 (2020), 779-794.
[100] Dube, T. and Stephen, D.N., On ideals of rings of continuous functions associated with sublocales, Topology and its Applications 284 (2020), 107360.
[101] Dube, T., Amenable and locally amenable algebraic frames, Order 37 (2020), 509-528.
[102] Dube, T., Characterizing realcompact locales via remainders, Georgian Mathematical Journal, vol. 28, no. 1, 2021, pp. 59-72.
[103] Dube, T. and Parsinia, M., On the Sum of z◦-Ideals in Two Classes of Subrings of C(X), Bulletin of the Iranian Mathematical Society 47, 1069–1080 (2021).
[104] Dube, T. and Taherifar, A., On the lattice of annihilator ideals and its applications, Communications in Algebra Vol. 49:6, 2021, 2444-2456.
[105] Dube, T. and Stephen, D.N., Mapping Ideals to Sublocales. Applied Categorical Structures Vol. 29, 747–772 (2021).
[106] Dube, T., Georgiou, D., Megaritis, A., Naidoo, I. and Sereti, F., Covering Dimension and Universality Property on Frames. Order 39, 187–208 (2022).
[107] Baboolal, D. and Ori, R.G. The Samuel Compactification and Uniform Coreflection of Nearness Frames, Proceedings of the Symposium on Categorical Topology, University of Cape Town, Cape Town, 17 - 20, 1994.
[108] Banaschewski, B., Another look at the localic Tychonoff Theorem, Commentationes Mathematicae Universitatis Carolinae, Vol. 29 (1988), No. 4, 647–656.
[109] Banaschewski, B., Lectures on Cauchy points of nearness frames, University of Cape Town, 1990.
[110] Banaschewski B., The frame envelope of a σ-frame, Quaestiones Mathematicae., 16(1):51–60, 1993.
[111] Banaschewski, B., Completion in pointfree topology, Lecture Notes in Mathematics & Applied Mathematics, SoCAT94, University of Cape Town, 1996.
[112] Banaschewski, B., Booleanization, Cahiers de Topologie et G´eom´etrie Diff´erentielle Cat´egoriques, tome 37, no.1 (1996), p.41-60.
[113] Banaschewski, B. and Gilmour, C., Pseudocompactness and the cozero part of a frame, Commentationes Mathematicae Universitatis Carolinae, Vol. 37 (1996), No. 3, 577–587.
[114] Banaschewski, B. and Gilmour, C., Realcompactness and the cozero part of a frame, Applied Categorical Structures 9 (2001), 395 - 417.
[115] Banaschewski, B. and Mulvey, C., Stone- ˇ Cech compactification of locales. I, Houston Journal of Mathematics 6 (1980), 301 - 312.
[116] Banaschewski, B and Pultr, A., Samuel compactification and completion of uniform frames, Mathematical Proceedings of the Cambridge Philosophical Society 108 (1990), 63 - 78.
[117] Banaschewski, B. and Pultr, A., Paracompactness Revisted, Applied Categorical Structures 1: 181-190,1993.
[118] Banaschewski, B. and Pultr, A., Cauchy points of uniform and nearness frames, Quaestiones Mathematicae, 19: 1 -2 (1996), 101 - 127.
[119] Banaschewski, B. and Pultr, A., A new look at pointfree metrization theorems, Commentationes Mathematicae Universitatis Carolinae, Vol.39 (1998), No.1, 167 -175.
[120] Banaschewski, B. and Pultr, A., Completion and Samuel compactification of nearness and uniform frames, Portugaliae Mathematica Vol. 69 (2012), No.2, pp. 113-126.
[121] Banaschewski, B., Hong, S.S. and Pultr, A., On the completion of nearness frames, Quaestiones Mathematicae, 21:1-2, 19-3, 1998.
[122] Bentley, H.L., Normal nearness spaces, Quaestiones Mathematicae, 2:1-3, 23-43 (1977).
[123] Bentley, H.L., Paracompact spaces, Topology and its Applications 39 (1991), 283-297.
[124] Bentley, H.L., and Hunsaker, W., ˇ Cech complete nearness spaces, Commentationes Mathematicae Universitatis Carolinae, Vol.33 (1992), No.2, 315 - 328.
[125] Borceux, F., Handbook of Categorical Algebra I, Basic Category Theory, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 1990.
[126] Chen, X., On binary coproducts of frames, Commentationes Mathematicae Universitatis Carolinae, Vol. 33(4) (1992), 699–712 .
[127] Dowker, C.H. and Papert, D., Quotient frames and subspaces, Proceedings of the London Mathematical Society (3) 16 (1966), 275 - 296.
[128] Fr´echet, M.M., Sur quelques points du calcul fonctionnel, Rendiconti del Circolo Matematico di Palermo (1884-1940) vol. 22, 1–72 (1906).
[129] Frith, J.L., Structured frames, PhD thesis, University of Cape Town, 1986.
[130] Frith, J.L., The category of uniform frames, Cahiers De Topologie Et G´eom´etrie Diff´erentielle Cat´egoriques, tome 31, no 4 (1990), p. 305-313.
[131] Frith, J.L. and Schauerte, A., A Method for Constructing Coreflections for Nearness Frames, Applied Categorical Structures (2014) 22:741–753.
[132] Haimene, M.N., An investigation of the strongness property for nearness frames, MSc mini-thesis, University of Namibia, 2018.
[133] Herrlich, H., A concept of nearness, General Topology and its Applications 5 (1974), 191 - 121,
[134] Herrlich, H. and Ori, R.G., Zero sets and complete regularity for nearness spaces, In: Categorical Topology and its Relations to Analysis, Algebra and Combinatorics, Prague, Czechoslovakia, 22 - 27 August 1988, Ed. Adamek, J. and MacLane, S.,World Scientific Publ. Co., Singapore, 1989, pp. 446-461.
[135] Hohti, A., On uniform paracompactness, Annales Academiae Scientiarum Fennicae Series A, I. Mathematica Dissertationes Vol. 36, 1981.
[136] Hong, S.S., Convergence in frames, Kyungpook Mathematical Journal 35 (1995), 85 - 93.
[137] Hong, S.S. and Kim, Y.K., Cauchy completions of nearness frames, Applied Categorical Structures, 3 (1995), pp. 371-377.
[138] Howes, N.R., Paracompactifications, preparacompactness and some problems of K. Morita and H. Tamano, Question and Answers in General Topology Vol. 10 (1992), 191-204.
[139] Isbell, J.R., Uniform Spaces, American Mathematical Society Mathematics Surveys 12, Providence, 1964.
[140] Isbell, J.R., Atomless parts of spaces, Mathematica Scandinavica 31 (1972), 5 - 32.
[141] Kˇr´ıˇz, I., A direct description of uniform completion in locales and a characterization of LT-groups, Cahiers de Topologie et G´eom´etrie Diff´erentielle Cat´egoriques, tome 27, no 1 (1986), p. 19-34.
[142] Kˇr´ıˇz, I. and Pultr, A., Peculiar behaviour of connected locales, Cahiers de Topologie et G´eom´etrie Diff´erentielle Cat´egoriques, tome 30, no 1 (1989), p. 25-43.
[143] Madden, J. and Vermeer, J., Lindel¨of locales and realcompactness, Mathematical Proceedings of the Cambridge Philosophical Society 99 (1986), 473 - 480.
[144] Marcus, N., Realcompactifications of frames, Master’s thesis, University of Cape Town, 1993.
[145] Marcus, N., Realcompactifications of frames, Commentationes Mathematicae Universitatis Carolinae, Vol.36(1995), No.2, 347–356.
[146] McKee, R.L., Zero-dimensional nearness spaces and extensions of topological spaces, Missouri Journal of Mathematical Sciences 6(2)(1994), 64–67.
[147] Michael, E., A note on paracompact spaces, Proceedings of the American Mathematical Society Vol. 8, No. 4 (1957), 822-828.
[148] Mugochi, M.M., Contributions to the theory of nearness in pointfree topology, PhD Thesis, University of South Africa, 2009.
[149] Mugochi, M.M., A short note on the role of grills in nearness frames, Hacettepe Journal of Mathematics and Statistics Vol. 41(1) (2012), 85–91.
[150] Naidoo, I., An interview with Themba Andrew Dube (A TAD Interview), Categories and General Algebraic Structures with Applications, Special Issue dedicated to Themba Dube, 2023.
[151] Naidoo, I., A note on precompact uniform frames, Topology and its Applications 153 (2005), 941–947.
[152] Naidoo, I., Strong Cauchy Completeness in Uniform Frames, Acta Mathematica Hungarica 116 (3), (2007), 273 - 284.
[153] Papert, D. and Papert, S., Sur les treillis des ouverts et les paratopologies, S´eminaire Ehresmann, Topologie Et G´eom´etrie Diff´erentielle, Tome 1 (1957-1958), 1 - 9.
[154] Picado, J. and Pultr, A., Frames and locales: Topology without points, Frontiers in Mathematics, Birkh¨auser/Springer Basel AG, Basel, 2012.
[155] Pultr, A., Pointless uniformities. I. Complete regularity, Commentationes Mathematicae Universitatis Carolinae, Vol. 25 (1984), No. 1, 91 - 104.
[156] Pultr, A., Pointless uniformities. II. (Dia)metrization, Commentationes Mathematicae Universitatis Carolinae, Vol. 25 (1984), No. 1, 105 - 120.
[157] Pultr, A., Remarks on metrizable locales, Rendiconti del Circolo Matematico di Palermo Series 2, 6 (1984), 247-158.
[158] Pultr, A. and Sichler, J., Frames: Topology without points, Lattice Theory: Special Topics and Applications: Volume 1, Chapter 2, 55 - 88, G. Gr¨atzer and F. Wehrung (eds.), Springer International Publishing Switzerland 2014.
[159] Reynolds, G., Alexandroff algebras and complete regularity, Proceedings of the American Mathematical Society 76, 322 - 326, 1979.
[160] Seo, J.R. and Lee, C.K., Categories of nearness frames, Communications of the Korean Mathematical Society 13 (1998), No. 4, pp. 847 - 854.
[161] Simmons, H., A framework for topology, Studies in Logic and the Foundations of Mathematics, Proceedings of the Wroclaw Logic Colloquium ’77, A. Macintyre, L. Pacholaki, J. Paris (eds.), North-Holland Publishing Company, 1978, 239-251.
[162] Simmons, H., Near-Discreteness of Modules and Spaces as measured by Gabriel and Cantor, Journal of pure and Applied Algebra 56 (1989), 119 -162.
[163] Simmons, H., An Introduction to Category Theory, Cambridge University Press, 2011.
[164] Sun, S.H., On paracompact locales and metric locales, Commentationes Mathematicae Universitatis Carolinae, Vol. 30 (1989), No. 1, 101 - 107.
[165] Townsend, C.F., Preframe Techniques in Constructive Locale Theory, PhD. Thesis, University of London, 1996.
[166] Tukey, J.W., Convergence and Uniformity in Topology, Annals of Mathematics Studies, Princeton Univ. Press, Princeton, NJ., 1940.
[167] Walters-Wayland, J.L., A Shirota Theorem for Frames, Applied Categorical Structures 7: 271 - 277, 1999.
[168] Weil, A., Sur les espaces `a structure uniforme et sur la topologie g´en´erale, Publications de L’Institut Math´ematique de L’Universit´e Strasbourg, Hermann & Cie, Paris 1937.