##
**The topology of elementary submodels.**
*(English)*
Zbl 0903.54002

This paper embarks on a somewhat unusual and quite interesting exploration of elementary submodels in topology. Perhaps the ultimate goal is the same (elementary submodels as an investigative tool in topology) but the more immediate goal is certainly different. As the authors remark, it is to “consider the operation of taking an elementary submodel to be yet another operation, like taking a subspace or an image under a nice map”. The thrust of this paper is to find examples of spaces \(X\) for which there exist elementary submodels \(M\) (suitably restricted) such that a naturally defined space \(X_M\) determined by \(X\) and \(M\) does or does not inherit properties of \(X\). The paper assumes some familiarity with the basics of elementary submodels but is otherwise widely accessible.

There are certainly too many results to list and no one of them is presented as the paper’s main result. To give the flavour however we mention just one. It is shown (2.17) that if \(X\) is Hausdorff sequential and \(M\) is a countably closed (meaning \(M^\omega \subset M\)) submodel (of a suitably large fragment) then \(X_M\) inherits from \(X\) any property that is preserved by both closed subsets and continuous images. For the expert reader, \(X_M\) is \(X\cap M\) endowed with the topology generated by \(\tau \cap M\) where \(\tau\) is the topology on \(X\). It is interesting to note that examples are provided to show that \(X_M\) need not be either a closed subset of \(X\) nor a continuous image of \(X\). The most interesting example in the paper is likely 7.20: an example of a compact space and a countably closed model \( M\) such that \(X_M\) is not normal. Anyone working with elementary submodels in topology will want to have a look at this paper.

There are certainly too many results to list and no one of them is presented as the paper’s main result. To give the flavour however we mention just one. It is shown (2.17) that if \(X\) is Hausdorff sequential and \(M\) is a countably closed (meaning \(M^\omega \subset M\)) submodel (of a suitably large fragment) then \(X_M\) inherits from \(X\) any property that is preserved by both closed subsets and continuous images. For the expert reader, \(X_M\) is \(X\cap M\) endowed with the topology generated by \(\tau \cap M\) where \(\tau\) is the topology on \(X\). It is interesting to note that examples are provided to show that \(X_M\) need not be either a closed subset of \(X\) nor a continuous image of \(X\). The most interesting example in the paper is likely 7.20: an example of a compact space and a countably closed model \( M\) such that \(X_M\) is not normal. Anyone working with elementary submodels in topology will want to have a look at this paper.

Reviewer: A.Dow (North York)

### MSC:

54A10 | Several topologies on one set (change of topology, comparison of topologies, lattices of topologies) |

03C62 | Models of arithmetic and set theory |

54D30 | Compactness |

54A25 | Cardinality properties (cardinal functions and inequalities, discrete subsets) |

54B99 | Basic constructions in general topology |

54D15 | Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.) |

54D55 | Sequential spaces |

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\textit{L. R. Junqueira} and \textit{F. D. Tall}, Topology Appl. 82, No. 1--3, 239--266 (1998; Zbl 0903.54002)

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