© 1996, Topology Atlas

In this brief note I will describe some situations in topology in which elementary submodels occur, hoping to convince the reader of its usefulness and consequently to take a look at the papers quoted throughout this note. The first complete and systematic exposition of this tecnique can be found in a paper of Alan Dow (4), in this paper it is seen that any application of elementary submodel is an application of the downward Lowenheim-Skolem theorem. For a different approach, relying only on the notion of a formula and a free variable, see the papers of Steve Watson (10), (9) and the forthcoming paper of Steve Watson and Alessandro Fedeli (7), in these papers a weak form of the Lowenheim-Skolem theorem is used. Some of the topics in which elementary submodels can be successfully used are:

1. Cardinal Functions: this is, perhaps, the best place to see how elementary submodels can be applied. Here any "closure argument" can be replaced by an application of the Lowenheim-Skolem theorem. See e.g. the proof of the well-known Arhangel'skii's result on the cardinality of Hausdorff spaces in (4), for more applications see (7),(9) and (10).

2. Function spaces, Corson-compact spaces: some interesting results in these topics have been obtained by Ingo Bandlow. In (2) it is shown a characterizationof Corson-compact spaces by means of countable elementary submodels. In (3) the author gives a characterization of the spaces C_p(X) for Corson-compact spaces.

3. Factorization theorems: these theorems are particularly important in dimension theory, for some results in this direction see (1) and (8).

4. Dyadic spaces: this is another field in which elementary submodels seem to be useful. See e.g. the elementary submodel proof of a generalization of the following result of Arhangel'skii and Ponomarev: "Every dyadic space of countable tightness is second countable" in (6).

Other applications which are of interest for topologists can be found in set-theory. One of the first application (a decomposition theorem for arbitrary partially ordered sets ) is due to Stevo Todorcevic (8). Basic results about sets (such as the delta-system lemma, the pressing-down lemma, etc.) in which elementary submodels are used in the proofs can be found in (4),(5) [see also (9)]. See (4) and (5) also for some applications of elementary submodels in forcing arguments. Certainly the above list is far to be complete but I hope there is enough in it to stimulate the interest of the reader.

(1) I. Bandlow, A construction in set-theoretic topology by means of elementary substructures, Zeitschr. f. math. Logik und Grundlagen d. Math. 37 (1991), 467-480

(2) I. Bandlow, A characterization of Corson-compact spaces, Comment.Math. Univ.Carolinae 32,3 (1991), 545-550

(3) I. Bandlow, On function spaces of Corson-compact spaces, Comment.Math. Univ.Carolinae 35,2 (1994), 347-356

(4) A. Dow, An introduction to applications of elementary submodels in topology, Top. Proc. 13 (1988), 17-72

(5) A. Dow, More set-theory for topologists, Top. Appl. 64 (1995), 243-300

(6) A. Fedeli and S. Watson, Continuous images of products of separable spaces, submitted

(7) A. Fedeli and S. Watson, Elementary submodels in topology, preprint

(8) S. Todorcevic, Directed sets and cofinal types, Trans. Amer. Math. Soc. 290 (1985), 711-723

(9) S. Watson, The construction of topological spaces: Planks and resolutions, in Recent Progress in General Topology (M. Husek and J. Van Mill eds.), Elsevier Science Publishers 1992, 675-757

(10) S. Watson, The Lindelof number of a power; an introduction to the use of elementary submodels in general topology, Top. Appl. 58 (1994), 25-34