An introduction to applications of elementary submodels to topology. (English) Zbl 0696.03024

For sets or classes M and N, M is said to be an elementary submodel of N if \(M\subseteq N\) and for all \(n\in \omega\) and formulas \(\phi\) with at most n free variables and all \(\{a_ 1,...,a_ n\}\subseteq M\), the formula \(\phi^ M(a_ 1,...,a_ n)\) holds iff the formula \(\phi^ N(a_ 1,...,a_ n)\) holds. Elementary submodels turned out to be a very powerful technical tool in set theoretic topology. The author validates this in the paper by proving some new and old theorems. He deals mainly with reflection problems of the following type: If \(\kappa\) is a cardinal and X is a space such that every subspace of X of cardinality at most \(\kappa\) has a property P, then, does this guarantee that X has P? To illustrate a wide spectrum of results obtained, let me quote only the most spectacular. (1) If every subspace of cardinality \(\omega_ 1\) of a countably compact space is metrizable, then the space itself is metrizable. Let \(k(X,P)=\inf \{| Y|:\) \(Y\subseteq X\) and Y does not have property \(P\}\). (2) If X is a space where every point has a neighborhood of cardinality at most \(\omega_ 1\), then the following holds: k(X, metrizable) \(\Rightarrow\) X metrizable iff there are no non- reflecting stationary sets. (3) Fleissner’s axiom R implies that there are no non-reflecting stationary sets. (4) If G is \(Fn(\omega_ 2,2)\)- generic over V, a model of CH, then in V[G], a first countable space of weight \(\omega_ 1\) is metrizable if each of its subspaces of cardinality at most \(\omega_ 1\) is metrizable.
Reviewer: A.Szymański


03E35 Consistency and independence results
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54A35 Consistency and independence results in general topology
54-02 Research exposition (monographs, survey articles) pertaining to general topology
03E55 Large cardinals
54E35 Metric spaces, metrizability