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A construction in set-theoretic topology by means of elementary substructures. (English) Zbl 0769.54013
A technique which is becoming increasingly explicit in general topology is the usage of elementary submodels of set-theory. A general approach is to take a topological space \(\langle X,\tau\rangle\) and a sufficiently large set \(H\supset X\) so that \(\langle H,\in\rangle\) is a model of all the properties of interest of the space. One then invokes the Löwenheim-Skolem-Tarski theorem to choose an \(M\subset H\) so that the structure \(\langle M,\in\rangle\) is an elementary submodel of \(\langle H,\in\rangle\) and such that each of the sets \(X\), \(\tau\) are elements of \(M\) (but neither in general is a subset). One might then study the properties of the much smaller sets \(\langle X\cap M, \tau\cap M\rangle\) and make such deductions as are possible about the original space. It is frequently the interaction between \(X\cap M\) and \(X\) that is most informative. In many cases however, \(X\cap M\) is in fact too small to be useful. The present paper shows that there is another natural smaller (than \(X\)) object, \(X(M)\), determined by \(X\) and \(M\), together with a mapping from \(X\) onto \(X(M)\), which can be usefully investigated.
Reviewer: A.Dow (North York)

MSC:
54B99 Basic constructions in general topology
03C55 Set-theoretic model theory
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