A few topological problems.

*(English)*Zbl 0688.54003This paper gives a broad list of unsolved problems, and a brief discussion of them. The problems concern Dowker filters, box products, metrizability in manifolds, small Dowker spaces, normality, Lindelöf, and shrinkability in products, \(C_ p(X)\), \(\theta\)-refinability, \(M_ 1\) vs \(M_ 3\), normal first countable spaces, and a few others. Many of the problems are known to be difficult.

{Some progress has been made on two of the problems: The problem about Dowker filters has been solved consistently by Z. Balogh and G. Gruenhage (they construct a model by iterated forcing where Dowker filters exist). The problem “can there be a closed discrete set of cardinality \(\aleph_ 1\) in a normal, first countable space which is not a \(G_{\delta}\)” has been answered “consistently yes” by S. Shelah [Irs. J. Math. 65, 219-224 (1989; see the following review)]. W. Fleissner [Proc. Am. Math. Soc. 46, 294-298 (1974; Zbl 0314.54028)] had proved earlier that the problem had a negative answer under \(V=L\); so the problem is independent of and consistent with ZFC).}

{Some progress has been made on two of the problems: The problem about Dowker filters has been solved consistently by Z. Balogh and G. Gruenhage (they construct a model by iterated forcing where Dowker filters exist). The problem “can there be a closed discrete set of cardinality \(\aleph_ 1\) in a normal, first countable space which is not a \(G_{\delta}\)” has been answered “consistently yes” by S. Shelah [Irs. J. Math. 65, 219-224 (1989; see the following review)]. W. Fleissner [Proc. Am. Math. Soc. 46, 294-298 (1974; Zbl 0314.54028)] had proved earlier that the problem had a negative answer under \(V=L\); so the problem is independent of and consistent with ZFC).}

Reviewer: J.E.Vaughan

##### MSC:

54-02 | Research exposition (monographs, survey articles) pertaining to general topology |

54A35 | Consistency and independence results in general topology |

54B10 | Product spaces in general topology |

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

54E20 | Stratifiable spaces, cosmic spaces, etc. |

54D20 | Noncompact covering properties (paracompact, Lindelöf, etc.) |

54E35 | Metric spaces, metrizability |

57P99 | Generalized manifolds |