Abstract
The \({G_\delta}\)-modification \({X_\delta}\) of a topological space X is the space on the same underlying set generated by, i.e. having as a basis, the collection of all \({G_\delta}\) subsets of X. Bella and Spadaro recently asked the following question(s): Is \({t(X_\delta) \le 2^{t(X)}}\) true for every (compact) T2 space X?
In this note we answer both questions: In the compact case affirmatively and in the non-compact case negatively. In the latter case we even show that it is consistent with ZFC that no upper bound exists for the tightness of the \({G_\delta}\)-modifications of countably tight, even Fréchet spaces.
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Acknowledgements
The second author thanks the support from the Mathematics Department of UNC Charlotte and in particular Professor Alan Dow. The fifth author thanks the hospitality of the Rényi Institute, most of the research on this paper was carried out during his visit there.
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Dedicated to the memory of Ákos Császár
In the research on and preparation of this paper the second, third, and fourth named authors were supported by NKFIH grant no. K 113047.
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Dow, A., Juhász, I., Soukup, L. et al. On the tightness of \({G_\delta}\)-modifications. Acta Math. Hungar. 158, 294–301 (2019). https://doi.org/10.1007/s10474-018-0864-1
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DOI: https://doi.org/10.1007/s10474-018-0864-1
Key words and phrases
- tightness
- Fréchet
- \({G_\delta}\)-modification
- compact
- Lindelöf
- pseudocharacter
- non-reflecting stationary set
- strongly compact cardinal