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Weakly Corson compact trees

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Abstract

We introduce and study a new topology on trees, that we call the countably coarse wedge topology. Such a topology is strictly finer than the coarse wedge topology and it turns every chain complete, rooted tree into a Fréchet–Urysohn, countably compact topological space. We show the rôle of such topology in the theory of weakly Corson and weakly Valdivia compacta. In particular, we give the first example of a compact space T whose every closed subspace is weakly Valdivia, yet T is not weakly Corson. This answers a question due to Ondřej Kalenda.

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Notes

  1. Since we won’t need the notion of retractional skeleton in our paper, we shall refer to [8] for its definition.

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Acknowledgements

The authors wish to express their gratitude to the anonymous referee for their insightful comments which substantially improved the paper.

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Author notes

  1. In memoriam: Benedetto Braggion (1948–2021).

Authors and Affiliations

  1. Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67, Prague 1, Czech Republic

    Tommaso Russo

  2. Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, 166 27, Prague 6, Czech Republic

    Tommaso Russo

  3. Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, Via Cesare Saldini 50, 20133, Milan, Italy

    Jacopo Somaglia

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  1. Tommaso Russo
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  2. Jacopo Somaglia
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Correspondence to Jacopo Somaglia.

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T. Russo was supported by the GAČR project 20-22230L; RVO: 67985840. J. Somaglia was supported by Università degli Studi di Milano. Both authors were also supported by Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of Istituto Nazionale di Alta Matematica (INdAM), Italy.

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Russo, T., Somaglia, J. Weakly Corson compact trees. Positivity 26, 33 (2022). https://doi.org/10.1007/s11117-022-00874-5

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