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Upwards preservation by elementary submodels. (English) Zbl 1002.54003

If a space \(X\) is a member of an elementary submodel \(M\) (of a suitable fragment) then a space \(X_M\) is defined where the base set for \(X_M\) is \(X\cap M\) and the topology for \(X_M\) is generated by the open subsets of \(X\) which are members of \(M\). This paper explores the extent to which properties of \(X_M\) will accurately predict properties of \(X\) (i.e. upward absoluteness) under quite mild assumptions on \(M\). Example 4.6 is a nice example showing that \(X_M\) can be uncountable separable while \(X\) itself need not be separable. A model \(M\) is said to be \(\omega\)-covering if every countable subset of \(M\) is a subset of a countable element of \(M\); there are \(\omega\)-covering models of size \(\omega_1\). Several examples show that the hypothesis of \(\omega\)-covering is sufficient to ensure upward absoluteness, for example a space being of pointwise countable type. However there is a nice discussion to show that \(\omega\)-covering is not sufficient for upward absoluteness of the Fréchet property, while the stronger \(\omega\)-closure property (\(M^\omega\subset M\)) does suffice. The paper closes with some discussion about preservation by \(\omega_1\)-closed forcing. In particular it is shown that if a first countable space fails to have a weak form of collectionwise Hausdorff (stationarily collectionwise Hausdorff), then the failure of this property is preserved by \(\omega_1\)-closed forcing.

MSC:

54A35 Consistency and independence results in general topology
03C62 Models of arithmetic and set theory
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54E35 Metric spaces, metrizability
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
54D30 Compactness
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
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