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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.

Reviewer: Alan Dow (Charlotte)

### 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.) |