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Some open categorical problems in Top. (English) Zbl 0791.54012

The paper gives a “guided tour” through 20 up-to-date open problems and 36 questions in Categorical Topology, with problems defined to be “wider” than questions. The majority of these problems and questions concern the category of topological spaces and its subcategories. All of them are understandable with basic knowledge of topological and categorical terminology, and they are accompanied by extensive references to the relevant literature.

MSC:

54B30 Categorical methods in general topology
54C35 Function spaces in general topology
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
54A05 Topological spaces and generalizations (closure spaces, etc.)
54D80 Special constructions of topological spaces (spaces of ultrafilters, etc.)
18B30 Categories of topological spaces and continuous mappings (MSC2010)
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[1] Adámek, J., H. Herrlich, and G.E. Strecker: 1990,Abstract and Concrete Categories, Wiley-Intersci. Publ., New York. · Zbl 0695.18001
[2] Adámek, J., J. Reitermann, and G.E. Strecker: 1985, ’Realization of cartesian closed topological hulls’,Manuscripta Math. 53, 1–33. · Zbl 0573.54006
[3] Adámek, J. and J. Rosický: 1988, ’Intersections of reflective subcategories’,Proc. Amer. Math. Soc. 103, 710–712. · Zbl 0675.18002
[4] Adámek, J. and J. Rosický: 1993, ’On injectivity classes in locally presented categories’,Trans. Amer. Math. Soc. (to appear). · Zbl 0789.18003
[5] Adámek, J., J. Rosický, and V. Trnková: 1990, ’Unexpected properties of locally presentable categories’,Alg. Univ. 27, 153–170. · Zbl 0701.18003
[6] Arens, R.F.: 1946, ’A topology for spaces of transformations’,Ann. Math. 47, 480–495. · Zbl 0060.39704
[7] Arhangel’skij, A.V. and R. Wiegandt: 1975, ’Connectedness and disconnectedness in topology’,Gen. Top. Appl. 5, 9–33. · Zbl 0329.54008
[8] Bentley, H.L. and H. Herrlich: 1992, ’Compactness = Completeness Total boundedness; a natural example of a non-reflective intersection of reflective subcategories’, in:Recent Developments of General Topology and its Applications (eds. W. Gähler, H. Herrlich, and G. Preuss), Akademic Verlag,67, 46–56. · Zbl 0790.54008
[9] Brandenburg, H.: 1983, ’An extension theorem for D-normal spaces’,Topology Appl. 15, 223–299. · Zbl 0506.54013
[10] Brandenburg, H. and M. Hušek: 1982,A Remark on Cartesian Closedness, Springer Lecture Notes Math.962, 33–38. · Zbl 0497.54008
[11] Brandenburg, H. and A. Mysior: 1984, ’For every Hausdorff spaceY there exists a non-trivial Moore space on which all continuous functions intoY are constant’,Pacific J. Math. 111, 1–8. · Zbl 0538.54016
[12] Breger, H.: 1971, ’Die Kategorie der kompakt erzeugten Räume als in Top coreflektive Kategorie mit Exponentialgesetz’, Diplomarbeit Univ. Heidelberg.
[13] Cagliari, F. and S. Mantovani: 1988, ’Localizations in universal topological categories’,Proc. Amer. Math. Soc. 103, 639–640. · Zbl 0652.54009
[14] Čech, E.: 1966,Topological Spaces (revised by Z. Frolík and M. Katětov), Prague.
[15] Choquet, G.: 1948, ’Convergences’,Ann. Univ. Grenoble; Sect. Sci. Math. Phys. 23, 57–112.
[16] Činčura, J.: 1985, ’Closed structures on categories of topological spaces’,Topology Appl. 20, 179–189. · Zbl 0568.54015
[17] Činčura, J.: 1990, ’Closed structures on reflective subcategories of the category of topological spaces’,Topology Appl. 37, 237–247. · Zbl 0724.18004
[18] Činčura, J.: 1991, ’Cartesian closed coreflective subcategories of the category of topological spaces’,Topology Appl. 41, 205–212. · Zbl 0755.18002
[19] Činčura, J.: 1992, ’Products in cartesian closed coreflective subcategories of the category of topological spaces’, Preprint.
[20] Dikranjan, D. and S. Watson: 1991, ’The category ofS({\(\alpha\)})-spaces is not co-wellpowered’, Preprint.
[21] Dow, A. and S. Watson: 1992, ’A subcategory of Top’,Trans. Amer. Math. Soc. (to appear). · Zbl 0823.54005
[22] Engelking, R.: 1989,General Topology, Heldermann Verlag, Berlin.
[23] Fischer, H.R.: 1959, ’Limesräume’,Math. Annalen 137, 269–303. · Zbl 0086.08803
[24] Freyd, P.J. and G.M. Kelly: 1972, ’Categories of continuous functors I’,J. Pure Appl. Algebra 2, 169–191. · Zbl 0257.18005
[25] Frič, R.: 1988, ’Remarks on sequential envelopes’,Rend. Instit. Matem. Univ. Trieste 20, 19–28. · Zbl 0696.54002
[26] Hajek, D.W. and A. Mysior: 1979, ’On non-simplicity of topological categories’,Springer Lecture Notes Math. 719, 84–93. · Zbl 0409.54014
[27] Heldermann, N.C.: 1980, ’The category ofD-completely regular spaces is simple’,Trans. Amer. Math. Soc. 262, 437–446. · Zbl 0459.54014
[28] Herrlich, H.: 1965, ’Wann sind alle stetigen Abbildungen inY konstant?’,Math. Zeitschr. 90, 152–154. · Zbl 0131.20402
[29] Herrlich, H.: 1967, ’Fortsetzbarkeit stetiger Abbildungen und Kompaktheitsgrad topologischer Räume’,Math. Zeitschr. 96, 64–72. · Zbl 0149.19501
[30] Herrlich, H.: 1969, ’On the concept of reflections in general topology’, in:Int. Symp. Extension Th. of Top. Structures and Appl., Berlin 1967, Akademie Verlag, 105–114.
[31] Herrlich, H.: 1974, ’A concept of nearness’,Gen. Top. Appl. 4, 191–212. · Zbl 0288.54004
[32] Herrlich, H.: 1975, ’Epireflective subcategories of Top need not be co-wellpowered’,Comment. Math. Univ. Carolin. 16, 713–716. · Zbl 0312.54004
[33] Herrlich, H.: 1987, ’Topological improvements of categories of structured sets’,Topology Appl. 27, 145–156. · Zbl 0632.54008
[34] Herrlich, H.: 1993a, ’Almost reflective subcategories of Top’,Topol. Appl. (to appear). · Zbl 0805.54016
[35] Herrlich, H.: 1993b, ’CompactT 0-spaces andT 0-compactifications’,Appl. Cat. Str. 1, 000-000.
[36] Herrlich, H. and M. Hušek: 1992, ’Categorical topology’, in:Recent Progress in General Topology (eds. M. Hušek and J. van Mill), Elsevier, Amsterdam, 369–403. · Zbl 0797.54023
[37] Herrlich, H. and G.E. Strecker: 1971, ’Algebra Topology = Compactness’,Gen. Top. Appl. 1, 283–287. · Zbl 0231.18007
[38] Hoffmann, R.E.: 1984, ’Co-wellpowered reflective subcategories’,Proc. Amer. Math. Soc. 90, 45–46. · Zbl 0531.18002
[39] Hong, S.S.: 1973, ’Onk-compactlike spaces and reflective subcategories’,Gen. Top. Appl. 3, 319–330. · Zbl 0272.54017
[40] Hušek, M.: 1969, ’The class ofk-compact spaces is simple’,Math. Zeitschr. 100, 123–126. · Zbl 0175.49601
[41] Hušek, M.: 1972a, ’Perfect images ofE-compact spaces’,Bull. Polish Acad. Sci. Math. 20, 41–45. · Zbl 0225.54019
[42] Hušek, M.: 1972b, ’Simple categories of topological spaces’, in:Proc. 3rd. Prague Top. Symp. 1971, Academia, Prague, 203–207.
[43] Hušek, M.: 1976,Lattices of Reflections and Coreflections in Continuous Structures, Springer Lecture Notes Math.504, 404–424.
[44] Hušek, M.: 1979a, ’Products of uniform spaces’,Czech. Math. J. 29, 130–141. · Zbl 0394.54014
[45] Hušek, M.: 1979b,Special classes of compact spaces, Springer Lecture Notes Math.719, 167–175.
[46] Hušek, M.: 1992, ’Čech–Stone-like compactifications of general topological spaces’,Comment. Math. Univ. Carolin. 33, 159–163. · Zbl 0754.54014
[47] Hušek, M. and J. van Mill (eds.): 1992,Recent Progress in General Topology, Elsevier, Amsterdam.
[48] Hušek, M., J. van Mill, and C. Mills: 1979, ’Some very small continua’, in:Top. Structure II, Math. Centre Tracts 115, 147–151.
[49] Hušek, M. and J. Pelant: 1974, ’Note about atom-categories of topological spaces’,Comment. Math. Univ. Carolin. 15, 767–773. · Zbl 0289.54004
[50] Hušek, M. and D. Pumplün: 1990, ’Disconnectedness’,Quaest. Math. 13, 449–459. · Zbl 0756.54010
[51] Hušek, M. and M.D. Rice: 1978, ’Productivity of coreflective subcategories of uniform spaces’,Gen. Top. Appl. 9, 295–306. · Zbl 0447.54009
[52] Hušek, M. and J. de Vries: 1987, ’Preservation of products by functors close to reflectors’,Topology Appl. 27, 171–189. · Zbl 0637.18002
[53] Kannan, V.: 1972, ’Reflexive cum coreflexive subcategories in topology’,Math. Ann. 195, 168–175. · Zbl 0215.09701
[54] Katětov, M.: 1965, ’On continuity structures and spaces of mappings’,Comment. Math. Univ. Carolin. 6, 257–278. · Zbl 0137.42003
[55] Keller, H.: 1968, ’Die Limes-Uniformisierbarkeit der Limesräume’,Math. Ann. 176, 334–341. · Zbl 0155.50302
[56] Kelly, G.M.: 1987, ’On the ordered set of reflective subcategories’,Bull. Austral. Math. Soc. 36, 137–152. · Zbl 0606.18001
[57] Kopperman, R.:Asymmetry and Duality in Topology (to appear). · Zbl 0858.54001
[58] Linton, F.E.I.: 1966, ’Some aspects of equational categories’, in:Proc. Conf. Cat. Algebra, La Jolla 1965 (Springer, Berlin), 84–94.
[59] Lowen-Colebunders, E. and R. Lowen: 1989, ’On the non-simplicity of some convergence categories’,Proc. Amer. Math. Soc. 105, 305–308. · Zbl 0679.18003
[60] Lowen-Colebunders, E. and Z.G. Szabo: 1990, ’On the simplicity of some categories of closure spaces’,Comment. Math. Univ. Carolin. 31, 95–98. · Zbl 0697.54007
[61] Makai, Jr. E.: 1991, ’Automorphisms and full embeddings of categories in algebra and topology’, in:Category Theory at Work (eds.: H. Herrlich and H.-E. Porst), Heldermann Verlag, 217–260. · Zbl 0748.18001
[62] Marjanovič, M.M. and A.R. Vučemilovič: 1985, ’Two non-homeomorphic countable spaces having homeomorphic squares’,Comment. Math. Univ. Carolin. 26, 579–588. · Zbl 0591.54006
[63] Van Mill, J. and M. Reed (eds.): 1990,Open Problems in Topology, Elsevier, Amsterdam.
[64] Mysior, A.: 1978, ’The category of all zerodimensional realcompact spaces is not simple’,Gen. Top. Appl. 8, 259–264. · Zbl 0404.54019
[65] Mysior, A.: 1980, ’Two remarks onD-regular spaces’,Glasnik Matem. 15, 153–156. · Zbl 0446.54021
[66] Nel, L.D.: 1977, ’Cartesian closed coreflective hulls’,Quaest. Math. 2, 269–283. · Zbl 0366.18008
[67] Orsati, A. and N. Rodino: 1986, ’Homeomorphisms between finite powers of topological spaces’,Topology Appl. 23, 271–277. · Zbl 0603.54009
[68] Pedicchio, M.C. and F. Rossi: 1985, ’A remark on monoidal closed structures on Top’,Rend. Circ. Mat. Palermo, II. Ser. Suppl. 11, 77–79. · Zbl 0637.18004
[69] Petz, D.: 1977, ’A characterization of a class of compact Hausdorff spaces’,Math. Hungarica 12, 407–408. · Zbl 0448.54023
[70] Richter, G.: 1982,Algebraic Categories of Topological Spaces, Springer Lecture Notes Math.962, 263–271. · Zbl 0508.54005
[71] Richter, G.: 1990, ’Characterizations of algebraic and varietal categories of topological spaces’,Seminarberichte Fernuniv. Hagen 36, 101–121.
[72] Rosický, J. and M. Sekanina: 1974, ’Realizations of topologies by set systems’, in:Coll. Math. J. Bolyai, Topics in Topology (Proc. Congr. Kestély 1972)8, 535–555.
[73] Schwarz, F.: 1982, ’Cartesian closedness, exponentiability, and final hulls in pseudotopological spaces’,Quaest. Math. 5, 289–304. · Zbl 0521.54005
[74] Schwarz, F.: 1984, ’Product compatible reflectors and exponentiability’, in:Proc. Int. Conf. Categ. Topology, Univ. of Toledo, Ohio, 1983, Heldermann Verlag, Berlin, 505–522.
[75] Smyth, M.B.: 1988, ’Stable local compactification I, Preprint.
[76] Tholen, W.: 1986, ’Prereflections and reflections’,Communications in Algebra 14, 717–740. · Zbl 0587.18002
[77] Tholen, W.: 1987, ’Reflective subcategories’,Topology Appl. 27, 201–212. · Zbl 0629.18004
[78] Tozzi, A.: 1986, ’US-spaces and closure operators’,Rend. Circolo Matem. Palermo, Suppl. 12, 291–300. · Zbl 0599.54016
[79] Trnková, V.: 1982a, ’Representation of commutative semigroups by products of topological spaces’, in:Gener. Topol. Rel. Mod. Anal. and Algebra (ed. J. Novák), Heldermann Verlag, 631–641.
[80] Trnková, V.: 1982b, ’Homeomorphisms of box-powers of space’,Glasnik Matem. 17, 131–137. · Zbl 0504.54006
[81] Trnková, V.: 1985, ’Countable Hausdorff spaces with countable weight’,Comment. Math. Univ. Carolin. 26, 749–770. · Zbl 0588.54012
[82] Trnková, V.: 1988, ’Homeomorphisms of products of subsets of Cantor discontinuum’,Dissertations Math. (Rozprawy Mat.) 268, 1–40. · Zbl 0659.54036
[83] Trnková, V., J. Adámek, and J. Rosický: 1990, ’Topological reflections revisited’,Proc. Amer. Math. Soc. 108, 605–612.
[84] Woods, R.G.: 1985, ’A class of extension properties that are not simply generated’,Topology Appl. 21, 287–295. · Zbl 0569.54025
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