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On resolvability, connectedness and pseudocompactness

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Abstract

We prove that: I. If L is a \(T_1\) space, \(|L|>1\) and \(d(L) \leq \kappa \geq \omega\), then there is a submaximal dense subspace X of \(L^{2^\kappa}\) such that \(|X|=\Delta(X)=\kappa\). II. If \(\mathfrak{c}\leq\kappa=\kappa^\omega<\lambda\) and \(2^\kappa=2^\lambda\), then there is a Tychonoff pseudocompact globally and locally connected space X such that \(|X|=\Delta(X)=\lambda\) and X is not \(\kappa^+\)-resolvable. III. If \(\omega_1\leq\kappa<\lambda\) and \(2^\kappa=2^\lambda\), then there is a regular space X such that \(|X|=\Delta(X)=\lambda\), all continuous real-valued functions on X are constant (so X is connected) and X is not \(\kappa^+\)-resolvable.

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References

  1. O. T. Alas, M. Sanchis, M. G. Tkacenko, V. V. Tkachuk and R. G. Wilson, Irresolvable and submaximal spaces: homogeneity versus σ-discreteness and new ZFC examples, Topology Appl., 107 (2000), 259–273.

  2. J. G. Ceder, On maximally resolvable spaces, Fund. Math., 55 (1964), 87–93.

  3. K. C. Ciesielski and J. Wojciechowski, Cardinality of regular spaces admitting only constant continuous functions, Topology Proc., 47 (2016), 313–329.

  4. W. W. Comfort and S. Garcia-Ferreira, Resolvability: a selective survey and some new results, Topology Appl., 74 (1996), 149–167.

  5. W. W. Comfort and Wanjun Hu, Resolvability properties via independent families, Topology Appl., 154 (2007), 205–214.

  6. W. W. Comfort and Wanjun Hu, Tychonoff expansions with prescribed resolvability properties, Topology Appl., 157 (2010), 839–856

  7. C. Costantini, On the resolvability of locally connected spaces, Proc. Amer. Math. Soc., 133 (2005), 1861–1864.

  8. R. Engelking, General Topology, Heldermann Verlag (Berlin, 1989).

  9. E. Hewitt, A problem in set theoretic topology, Duke Math. J., 10 (1943), 309–333.

  10. I. Juhász, L. Soukup and Z. Szentmikl´ossy, Coloring Cantor sets and resolvability of pseudocompact spaces, Comment. Math. Univ. Carolin., 59 (2018), 523–529.

  11. I. Juhász, L. Soukup and Z. Szentmikl´ossy, D-forced spaces: a new approach to resolvability, Topology Appl., 153 (2006), 1800–1824.

  12. I. Juhász and J. van Mill, Nowhere constant families of maps and resolvability, arXiv:2312.12257 (2023).

  13. K. Kunen, A. Szymanski and F. Tall, Baire irresolvable spaces and ideal theory, Ann. Math. Syleziana, 2 (1986), 98–107.

  14. K. Kunen and F. Tall, On the consistency of the non-existence of Baire irresolvable spaces, http://at.yorku.ca/v/a/a/a/27.htm (1998).

  15. A. E. Lipin, Resolvability and complete accumulation points, Acta Math. Hungar., 170 (2023), 661–669.

  16. V. I. Malykhin, Borel resolvability of compact spaces and their subspaces, Math. Notes, 64 (1998), 607–615.

  17. J. van Mill, Every crowded pseudocompact ccc space is resolvable, Topology Appl., 213 (2016), 127–134.

  18. Y. F. Ortiz-Castillo and A. H. Tomita, Pseudocompactness and resolvability, Fund. Math., 241 (2018), 127–142

  19. K. Padmavally, An example of a connected irresolvable Hausdorff space, Duke Math. J., 20 (1953), 513–520.

  20. O. Pavlov, On resolvability of topological spaces, Topology Appl., 126 (2002), 37–47.

  21. O. Pavlov, Problems on (ir)resolvability, in: Open Problems in Topology II, Elsevier B.V. (2007).

  22. E. G. Pytkeev, Resolvability of countably compact regular spaces, Proc. Steklov Inst. Math., Algebra. Topology. Mathematical Analysis (2002), S152–S154.

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Acknowledgement

The author is grateful to Maria A. Filatova for attention to this work and to the referee for useful remarks.

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  1. Krasovskii Institute of Mathematics and Mechanics, Ural Federal University, Yekaterinburg, Russia

    A. E. Lipin

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Lipin, A.E. On resolvability, connectedness and pseudocompactness. Acta Math. Hungar. 172, 519–528 (2024). https://doi.org/10.1007/s10474-024-01423-0

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