Convergence-theoretic characterizations of compactness. (English) Zbl 1022.54015
The author characterizes a variety of notions related to compactness in terms of concretely reflective convergence subcategories: topologies, paratopologies, hypotopologies and pseudotopologies. He also gives characterizations of hyperquotient maps (perfect, quasi-perfect, adherent and closed) and quotient maps (quotient, hereditarily quotient, countably bi-quotient, biquotient and almost open) in terms of various degrees of compactness of their fiber relations, and of sundry relaxations of inverse continuity. The results require too much notation to be describe here, but a good introduction to the theory and well arranged preliminaries are given.
Reviewer: Haruto Ohta (Ohya / Shizuoka)
MSC:
| 54D60 | Realcompactness and realcompactification |
| 54A20 | Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.) |
| 54B30 | Categorical methods in general topology |
| 54D30 | Compactness |
| 54C10 | Special maps on topological spaces (open, closed, perfect, etc.) |
Keywords:
compactness; convergence; pseudotopology; pretopology; quotient map; perfect map; compactoid filter; hyperquotient mapsReferences:
| [1] | Adámek, J.; Herrlich, H.; Strecker, E., Abstract and Concrete Categories (1990), Wiley: Wiley New York · Zbl 0695.18001 |
| [2] | Arhangel’skii, A. V., Bisequential spaces, tightness of products, and metrizability conditions in topological groups, Trans. Moscow Math. Soc., 55, 207-219 (1994) · Zbl 0842.54004 |
| [3] | Choquet, G., Convergences, Ann. Univ. Grenoble, 23, 55-112 (1947-1948) · Zbl 0031.28101 |
| [4] | Day, B. J.; Kelly, G. M., On topological quotient maps preserved by pullbacks or products, Proc. Cambridge Philos. Soc., 67, 553-558 (1970) · Zbl 0191.20801 |
| [5] | Dolecki, S., Lower semicontinuity of marginal functions, (Selected Topics in Operations Research and Mathematical Economics (1984), Springer: Springer Berlin), 30-41 · Zbl 0547.54013 |
| [6] | Dolecki, S., Convergence-theoretic methods in quotient quest, Topology Appl., 73, 1-21 (1996) · Zbl 0862.54001 |
| [7] | Dolecki, S., Active boundaries of upper semicontinuous and compactoid relations; closed and inductively perfect maps, Rostock Math. Coll., 54, 51-68 (2000) · Zbl 0961.54004 |
| [8] | Dolecki, S.; Greco, G. H., Familles pseudotopologiques et compacité, C. R. Acad. Sci. Paris, 296, 211-214 (1983) · Zbl 0526.54001 |
| [9] | Dolecki, S.; Greco, G. H.; Lechicki, A., Compactoid and compact filters, Pacific J. Math., 117, 69-98 (1985) · Zbl 0511.54012 |
| [10] | Dolecki, S.; Greco, G. H.; Lechicki, A., When do the upper Kuratowski topology (homeomorphically, Scott topology) and the cocompact topology coincide?, Trans. Amer. Math. Soc., 347, 2869-2884 (1995) · Zbl 0845.54005 |
| [11] | Dolecki, S.; Lechicki, A., Semi-continuité supérieure forte et filtres adhérents, C. R. Acad. Sci. Paris, 293, 219-221 (1981) · Zbl 0486.54003 |
| [12] | Dolecki, S.; Mynard, F., Convergence-theoretic mechanisms behind product theorems, Topology Appl., 104, 67-99 (2000) · Zbl 0953.54002 |
| [13] | Engelking, R., Topology (1989), Heldermann Verlag: Heldermann Verlag Berlin · Zbl 0684.54001 |
| [14] | Kac, M. P., Characterization of some classes of pseudotopological linear spaces, (Convergence Structures and Applications to Analysis (1980), Akademie-Verlag) · Zbl 0457.46011 |
| [15] | Kent, D. C., Convergence quotient maps, Fund. Math., 65, 197-205 (1969) · Zbl 0179.51002 |
| [16] | Kuratowski, K., Topology (1966), Academic Press/PWN: Academic Press/PWN New York/Warsaw · Zbl 0158.40802 |
| [17] | Michael, E., Complete spaces and tri-quotient maps, Illinois J. Math., 21, 716-733 (1971) · Zbl 0386.54007 |
| [18] | Michael, E., Inductively perfect maps and tri-quotient maps, Proc. Amer. Math. Soc., 82, 115-119 (1981) · Zbl 0463.54013 |
| [19] | Ostrovsky, A. V., Triquotient and inductively perfect maps, Topology Appl., 23, 25-28 (1986) · Zbl 0591.54010 |
| [20] | Penot, J. P., Compact nets, filters and relations, J. Math. Anal. Appl., 93, 400-417 (1983) · Zbl 0537.54002 |
| [21] | Pettis, B. J., Cluster sets of nets, Proc. Amer. Math. Soc., 22, 386-391 (1969) · Zbl 0181.25601 |
| [22] | Pillot, M., Tri-quotient maps become inductively perfect with the aid of consonance and continuous selections, Topology Appl., 104, 237-254 (2000) · Zbl 0947.54007 |
| [23] | Schroder, M., Solid convergences spaces, Bull. Australian Math. Soc., 8, 443-459 (1973) · Zbl 0254.54003 |
| [24] | Smithson, R. E., Subcontinuity for multifunctions, Pacific J. Math., 61, 283-288 (1975) · Zbl 0317.54022 |
| [25] | Topsøe, F., Compactness in spaces of measures, Studia Math., 36, 195-212 (1970) · Zbl 0201.06202 |
| [26] | Urysohn, P., Sur les classes (L) de M. Fréchet, Ens. Math., 25, 77-83 (1926) · JFM 52.0582.03 |
| [27] | Uspenskij, V. V., Tri-quotient maps are preserved by infinite products, Proc. Amer. Math. Soc., 123, 3567-3574 (1995) · Zbl 0845.54010 |
| [28] | Vaughan, J. E., Products of topological spaces, Gen. Topology Appl., 8, 207-217 (1978) · Zbl 0396.54016 |
| [29] | Wilker, P., Adjoint products and hom functors in general topology, Pacific J. Math., 34, 269-283 (1970) · Zbl 0205.52703 |
| [30] | Wyler, O., Solid convergence structure = pseudotopology, Bull. Australian Math. Soc., 15, 273-275 (1976) · Zbl 0328.54002 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
