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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
##### Keywords:
elementary submodel
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