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\({G_\delta}\) covers of compact spaces

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Abstract

We solve a long standing question due to Arhangel’skii by constructing a compact space which has a \({G_\delta}\) cover with no continuum-sized (\({G_\delta}\))-dense subcollection. We also prove that in a countably compact weakly Lindelöf normal space of countable tightness, every \({G_\delta}\) cover has a \({\mathfrak{c}}\)-sized subcollection with a \({G_\delta}\)-dense union and that in a Lindelöf space with a base of multiplicity continuum, every \({G_\delta}\) cover has a continuum sized subcover. We finally apply our results to obtain a bound on the cardinality of homogeneous spaces which refines De la Vega’s celebrated theorem on the cardinality of homogeneous compacta of countable tightness.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Catania, Cittá universitaria, viale A. Doria 6, 95125, Catania, Italy

    S. Spadaro

  2. Department of Mathematics, Faculty of Science and Engineering, York University, Toronto, ON, M3J 1P3, Canada

    P. Szeptycki

Authors
  1. S. Spadaro
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  2. P. Szeptycki
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Corresponding author

Correspondence to S. Spadaro.

Additional information

The first-named author is grateful to FAPESP for financial support through postdoctoral grant 2013/14640-1, Discrete sets and cardinal invariants in set-theoretic topology and to Ofelia Alas for useful discussion.

The second-named author acknowledges support from NSERC grant 238944.

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Spadaro, S., Szeptycki, P. \({G_\delta}\) covers of compact spaces. Acta Math. Hungar. 154, 252–263 (2018). https://doi.org/10.1007/s10474-017-0785-4

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  • DOI: https://doi.org/10.1007/s10474-017-0785-4

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