A few topological problems. (English) Zbl 0688.54003

Rudin, Mary Ellen

Commentat. Math. Univ. Carol. 29, No. 4, 743-746 (1988).

This 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).}

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

Keywords:

Problems; box products; metrizability in manifolds; small Dowker spaces; normality; Lindelöf; shrinkability in products; \(\theta\)-refinability; \(M_ 1\); \(M_ 3\); normal first countable spaces; Dowker filters

Citations:

Zbl 0688.54004; Zbl 0314.54028