In mathematics, a Menger space is a topological space that satisfies a certain basic selection principle that generalizes σ-compactness. A Menger space is a space in which for every sequence of open covers of the space there are finite sets such that the family covers the space.

History

edit

In 1924, Karl Menger [1] introduced the following basis property for metric spaces: Every basis of the topology contains a countable family of sets with vanishing diameters that covers the space. Soon thereafter, Witold Hurewicz [2] observed that Menger's basis property can be reformulated to the above form using sequences of open covers.

Menger's conjecture

edit

Menger conjectured that in ZFC every Menger metric space is σ-compact. A. W. Miller and D. H. Fremlin[3] proved that Menger's conjecture is false, by showing that there is, in ZFC, a set of real numbers that is Menger but not σ-compact. The Fremlin-Miller proof was dichotomic, and the set witnessing the failure of the conjecture heavily depends on whether a certain (undecidable) axiom holds or not.

Bartoszyński and Tsaban [4] gave a uniform ZFC example of a Menger subset of the real line that is not σ-compact.

Combinatorial characterization

edit

For subsets of the real line, the Menger property can be characterized using continuous functions into the Baire space  . For functions  , write   if   for all but finitely many natural numbers  . A subset   of   is dominating if for each function   there is a function   such that  . Hurewicz proved that a subset of the real line is Menger iff every continuous image of that space into the Baire space is not dominating. In particular, every subset of the real line of cardinality less than the dominating number   is Menger.

The cardinality of Bartoszyński and Tsaban's counter-example to Menger's conjecture is  .

Properties

edit
  • Every compact, and even σ-compact, space is Menger.
  • Every Menger space is a Lindelöf space
  • Continuous image of a Menger space is Menger
  • The Menger property is closed under taking   subsets
  • Menger's property characterizes filters whose Mathias forcing notion does not add dominating functions.[5]

References

edit
  1. ^ Menger, Karl (1924). "Einige Überdeckungssätze der Punktmengenlehre". Selecta Mathematica. Vol. 133. pp. 421–444. doi:10.1007/978-3-7091-6110-4_14. ISBN 978-3-7091-7282-7. {{cite book}}: |journal= ignored (help)
  2. ^ Hurewicz, Witold (1926). "Über eine verallgemeinerung des Borelschen Theorems". Mathematische Zeitschrift. 24 (1): 401–421. doi:10.1007/bf01216792. S2CID 119867793.
  3. ^ Fremlin, David; Miller, Arnold (1988). "On some properties of Hurewicz, Menger and Rothberger" (PDF). Fundamenta Mathematicae. 129: 17–33. doi:10.4064/fm-129-1-17-33.
  4. ^ Bartoszyński, Tomek; Tsaban, Boaz (2006). "Hereditary topological diagonalizations and the Menger–Hurewicz Conjectures". Proceedings of the American Mathematical Society. 134 (2): 605–615. arXiv:math/0208224. doi:10.1090/s0002-9939-05-07997-9. S2CID 9931601.
  5. ^ Chodounský, David; Repovš, Dušan; Zdomskyy, Lyubomyr (2015-12-01). "Mathias Forcing and Combinatorial Covering Properties of Filters". The Journal of Symbolic Logic. 80 (4): 1398–1410. arXiv:1401.2283. doi:10.1017/jsl.2014.73. ISSN 0022-4812. S2CID 15867466.