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Zbl 0696.03024
Dow, Alan
An introduction to applications of elementary submodels to topology. (English)
[J] Topology Proc. 13, No.1, 17-72 (1988). ISSN 0146-4124

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 $\{a\sb 1,...,a\sb n\}\subseteq M$, the formula $\phi\sp M(a\sb 1,...,a\sb n)$ holds iff the formula $\phi\sp N(a\sb 1,...,a\sb n)$ 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 $\omega\sb 1$ of a countably compact space is metrizable, then the space itself is metrizable. Let $k(X,P)=\inf \{\vert Y\vert:$ $Y\subseteq X$ and Y does not have property $P\}$. (2) If X is a space where every point has a neighborhood of cardinality at most $\omega\sb 1$, 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 $Fn(\omega\sb 2,2)$- generic over V, a model of CH, then in V[G], a first countable space of weight $\omega\sb 1$ is metrizable if each of its subspaces of cardinality at most $\omega\sb 1$ is metrizable.
[A.Szymański]

MSC 2000:: *03E35 Consistency and independence results (set theory)
54A25 Cardinality properties of topological spaces
54A35 Consistency and independence results (general topology)
54-02 Research monographs (general topology)
03E55 Large cardinals
54E35 Metric spaces, metrizability

Keywords: metrizability; cardinality properties of subspaces; countable tightness; elementary submodel; reflection; Fleissner's axiom; non-reflecting stationary sets

Cited in: Zbl 1028.54027

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