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The Combinatorics of Open Covers

Abstract

The combinatorics of open covers is a study of Cantor’s diagonal argument in various contexts. The field has its roots in a few basic selection principles that arose from the study of problems in analysis, dimension theory, topology and set theory. The reader will also find that some familiar works are appearing in new clothes in our survey. This is particularly the case in connection with such problems as determining the structure of compact scattered spaces and a number of classical problems in topology. We hope that the new perspective in which some of these classical enterprises are presented will lead to further progress. In this article we also attempt to give the reader an overview of the problems and techniques that are currently fueling much of the rapidly increasing current activity in the combinatorics of open covers.

Keywords

  • Topological Space
  • Open Cover
  • Winning Strategy
  • Metrizable Space
  • Selection Principle

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Fig. 1

Notes

  1. 1.

    Strictly speaking, the definition we give here equivalent to but not he original one given by Alexandroff. This will be clarified when Alexandroff’s original definition is given below in Sect. 6.

  2. 2.

    The operation \(\oplus \) on \(^{\kappa }2\) is coordinate-wise addition modulo 2.

  3. 3.

    That is, the topology induced by the metric \(p\) on \([0,\, 1]^{R}\), where \(p(f, g) := \sup \{\vert f(y) -g(y)\vert : y\in R\}\).

  4. 4.

    See [174], Proposition 8.

  5. 5.

    We use the following abbreviations: CC = Corson compact, EC = Eberlein compact, SEC = strong Eberlein compact, RNC = Radon-Nikodým compact.

  6. 6.

    We define the notion of a Hurewicz space later. These are spaces with a certain covering property.

  7. 7.

    This covering property is investigated in [186].

  8. 8.

    This covering property is a cozero set version of \((\alpha _1)\) in Kočinac [108].

  9. 9.

    This was first given for strongly zero-dimensional perfectly normal spaces \(X\) in the first version of Tsaban and Zdomskyy [189].

  10. 10.

    “metrizable”can be replaced by “perfectly normal” [91].

  11. 11.

    “R” for Rothberger. Rothberger did not consider \(\mathsf{S}_1(\mathfrak {D},\mathfrak {D})\). It has become customary to use the “R” in connection with selection principles of the form \(\mathsf{S}_1(\fancyscript{A},\fancyscript{A})\) since Rothberger introduced the first prototype of such a selection principle.

  12. 12.

    This was first proved for separable metrizable spaces in [161].

  13. 13.

    We are taking the liberty of giving an equivalent reformulation of Menger’s original conjecture. Menger did not mention \(\mathsf{S}_{fin}({\fancyscript{O}},{\fancyscript{O}})\) at all.

  14. 14.

    Note that this example is not a subspace of the real line.

  15. 15.

    “M” here refers to Menger. It has become common to call selection principles of the form \(\mathsf{S}_{fin}({\fancyscript{A}},{\fancyscript{B}})\) an “M-property”, with reference to the Menger property \(\mathsf{S}_{fin}({\fancyscript{O}},{\fancyscript{O}})\) which was the first prototype of this type of selection principle. Note that Hurewicz actually introduced this prototype, and poved that in metrizable spaces it is equivalent to a basis property introduced by Menger.

  16. 16.

    This was first proved for separable metrizable spaces in [161].

  17. 17.

    Note that the definition for weakly infinite dimensionality given here is the original definition, while the definition we gave earlier in Sect. 3 is an equivalent form of the original definition.

  18. 18.

    \({\mathbb K}\) is the well-known example of R. Pol [134] which answered Alexandroff’s problem.

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Sakai, M., Scheepers, M. (2014). The Combinatorics of Open Covers. In: Hart, K., van Mill, J., Simon, P. (eds) Recent Progress in General Topology III. Atlantis Press, Paris. https://doi.org/10.2991/978-94-6239-024-9_18

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