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The topology of elementary submodels. (English) Zbl 0903.54002

This paper embarks on a somewhat unusual and quite interesting exploration of elementary submodels in topology. Perhaps the ultimate goal is the same (elementary submodels as an investigative tool in topology) but the more immediate goal is certainly different. As the authors remark, it is to “consider the operation of taking an elementary submodel to be yet another operation, like taking a subspace or an image under a nice map”. The thrust of this paper is to find examples of spaces \(X\) for which there exist elementary submodels \(M\) (suitably restricted) such that a naturally defined space \(X_M\) determined by \(X\) and \(M\) does or does not inherit properties of \(X\). The paper assumes some familiarity with the basics of elementary submodels but is otherwise widely accessible.
There are certainly too many results to list and no one of them is presented as the paper’s main result. To give the flavour however we mention just one. It is shown (2.17) that if \(X\) is Hausdorff sequential and \(M\) is a countably closed (meaning \(M^\omega \subset M\)) submodel (of a suitably large fragment) then \(X_M\) inherits from \(X\) any property that is preserved by both closed subsets and continuous images. For the expert reader, \(X_M\) is \(X\cap M\) endowed with the topology generated by \(\tau \cap M\) where \(\tau\) is the topology on \(X\). It is interesting to note that examples are provided to show that \(X_M\) need not be either a closed subset of \(X\) nor a continuous image of \(X\). The most interesting example in the paper is likely 7.20: an example of a compact space and a countably closed model \( M\) such that \(X_M\) is not normal. Anyone working with elementary submodels in topology will want to have a look at this paper.
Reviewer: A.Dow (North York)

MSC:

54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
03C62 Models of arithmetic and set theory
54D30 Compactness
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54B99 Basic constructions in general topology
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
54D55 Sequential spaces
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[1] Alster, K.; Przymusiński, T., Normality and Martin’s Axiom, Fund. Math., 91, 123-131 (1976) · Zbl 0334.54016
[2] Arhangel’skiĭ, A. V., Bicompact sets and the topology of spaces, Trans. Moscow Math. Soc., 13, 1-62 (1965) · Zbl 0162.26602
[3] Arhangel’skiĭ, A. V., A characterization of very \(k\)-spaces, Czechoslovak Math. J., 18, 392-395 (1968) · Zbl 0165.25601
[4] Balogh, Z., On compact Hausdorff spaces of countable tightness, (Proc. Amer. Math. Soc., 105 (1989)), 755-764 · Zbl 0687.54006
[5] Balogh, Z., On collectionwise normality of locally compact, normal spaces, Trans. Amer. Math. Soc., 323, 389-411 (1991) · Zbl 0736.54017
[6] Balogh, Z., A small Dowker space in ZFC, (Proc. Amer. Math. Soc., 124 (1996)), 2555-2560 · Zbl 0876.54016
[7] Bandlow, I., A construction in set-theoretic topology by means of elementary substructures, Z. Math. Logik Grundlag. Math., 37, 467-480 (1991) · Zbl 0769.54013
[8] Bing, R. H., Metrization of topological spaces, Canad. J. Math., 3, 175-186 (1951) · Zbl 0042.41301
[9] Charlesworth, A.; Hodel, R.; Tall, F. D., On a theorem of Jones and Heath concerning separable normal spaces, (Colloq. Math., 34 (1975)), 33-37 · Zbl 0312.54006
[10] Daniels, P., Normal \(k\)’-spaces can be collectionwise normal, Fund. Math., 138, 225-234 (1991) · Zbl 0738.54001
[11] Dow, A., An introduction to applications of elementary submodels to topology, (Topology Proc., 13 (1988)), 17-72 · Zbl 0696.03024
[12] Dow, A., Set theory in topology, (Hušek, M.; van Mill, J., Recent Progress in General Topology (1992), North-Holland: North-Holland Amsterdam), 167-197 · Zbl 0796.54001
[13] Dow, A., More set theory for topologists, Topology Appl., 64, 243-300 (1995) · Zbl 0837.54001
[14] Engelking, R., General Topology (1989), Heldermann · Zbl 0684.54001
[15] Fedorčuk, V. V., Fully closed mappings and the consistency of some theorems of general topology with the axioms of set theory, Math. USSR-Sb., 28, 1-26 (1976)
[16] Fleissner, W. G., Theorems from measure axioms, counterexamples from ◊\(^{++}\), (Tall, F. D., The Work of Mary Ellen Rudin. The Work of Mary Ellen Rudin, Ann. New York Acad. Sci., 705 (1993), New York Acad. Sci: New York Acad. Sci New York), 67-77 · Zbl 0830.54003
[17] Grunberg, R.; Junqueira, L. R.; Tall, F. D., Forcing and normality, Topology Appl., 84 (1998) · Zbl 0923.54020
[18] Hechler, S. H., Generalization of almost disjointness, \(c\)-sets, and the Baire number of βN  \(N\), Gen. Topology Appl., 8, 93-110 (1978)
[19] Ismail, M.; Nyikos, P., On spaces in which countably compact sets are closed, and hereditary properties, Topology Appl., 11, 281-292 (1980) · Zbl 0434.54018
[20] Juhász, I., Cardinal functions in topology: ten years later, (Math. Center Tracts (1980)) · Zbl 0479.54001
[21] Nyikos, P. J.; Shapirovskiĭ, B.; Szentmiklóssy, Z.; Veličković, B., Complete normality and countable compactness, (Topology Proc., 17 (1992)), 395-403 · Zbl 0792.54021
[22] Tall, F. D., Set-theoretic consistency results and topological theorems concerning the normal Moore space conjecture and related problems, Ph.D. Thesis (1969), Madison · Zbl 0358.54011
[23] Tall, F. D., More topological applications of generic huge embeddings, Topology Appl., 44, 353-358 (1992) · Zbl 0805.54007
[24] Warren, N. M., Properties of Stone-Čech compactifications of discrete spaces, (Proc. Amer. Math. Soc., 33 (1972)), 599-606 · Zbl 0241.54016
[25] Watson, S., The Lindelöf number of a power; an introduction to the use of elementary submodels in general topology, Topology Appl., 58, 25-34 (1994) · Zbl 0836.54004
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