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On regular separable countably compact ℝ-rigid spaces

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Abstract

A topological space X is said to be Y-rigid if any continuous map f: X → Y is constant. In this paper we construct a number of examples of regular countably compact ℝ-rigid spaces with additional properties like separability and first countability. This way we answer several questions of Tzannes, Banakh, Ravsky, as well as get a consistent example of ℝ-rigid Nyikos space. Also, we show that it is consistent with ZFC that for every cardinal κ < c there exists a regular separable countably compact space X which is Y-rigid with respect to any T1 space Y of pseudocharacter ≤ κ.

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Acknowledgements

The authors acknowledge the Referee for numerous valuable remarks, in particular, for the idea of generalizing Lemma 4.2 which significantly improves Theorem 4.3, and Piotr Borodulin-Nadzieja for his comments on Question 4. In particular, Piotr convinced us that this question cannot be solved using tight gaps.

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Authors and Affiliations

  1. Institut für Mathematik, Kurt Gödel Research Center, Universität Wien, Kolingasse 14-16, 1090, Wien, Austria

    Serhii Bardyla & Lyubomyr Zdomskyy

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  1. Serhii Bardyla
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  2. Lyubomyr Zdomskyy
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Correspondence to Serhii Bardyla.

Additional information

The first author was supported by the Austrian Science Fund FWF (grants I 3709-N35 and M 2967).

The second author was supported by the Austrian Science Fund FWF (grants I 3709-N35 and I 2374-N35).

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Bardyla, S., Zdomskyy, L. On regular separable countably compact ℝ-rigid spaces. Isr. J. Math. 255, 783–810 (2023). https://doi.org/10.1007/s11856-022-2454-8

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  • DOI: https://doi.org/10.1007/s11856-022-2454-8

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