Strong Fréchet properties of spaces constructed from squares and AD families

William Chen-Mertens; César Corral-Rojas; Paul J. Szeptycki

doi:10.4995/agt.2023.18504

Authors

William Chen-Mertens York University https://orcid.org/0000-0001-6611-878X
César Corral-Rojas York University https://orcid.org/0000-0003-1865-7767
Paul J. Szeptycki York University https://orcid.org/0009-0008-0960-5811

DOI:

https://doi.org/10.4995/agt.2023.18504

Keywords:

Fréchet-Urysoh, bi-sequential, α1-space, W-space, w-space

Abstract

We answer questions of Arhangel'skiĭ using spaces defined from combinatorial objects. We first establish further convergence properties of a space constructed from □ ( κ ) showing it is Fréchet-Urysohn for finite sets and a w-space that is not a W-space. We also show that under additional assumptions it may be not bi-sequential, and so providing a consistent example of an absolutely Fréchet α₁ space that is not bisequential. In addition, if we do not require the space being α₁, we can construct a ZFC example of a countable absolutely Fréchet space that is not bisequential from an almost disjoint family of subsets of the natural numbers.

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References

A. V. Arkhangel'skiĭ, The spectrum of frequencies of a topological space and the product operation, Trans. Moscow Math. Soc. 40, no. 2 (1981), 163-200.

A. V. Arkhangel'skiĭ and A. Vladimirovich, The frequency spectrum of a topological space and classification of spaces, Doklady Akademii Nauk 206, no. 2 (1972), 265-268.

B. Balcar, J. Dočkálková and P. Simon, Almost disjoint families of countable sets, Finite and infinite sets, North-Holland (1984), 59-88. https://doi.org/10.1016/B978-0-444-86893-0.50009-7

W. Chen-Mertens and P. J. Szeptycki, The effect of forcing axioms on the tightness of the ${G}_delta$-modification, Fundamenta Mathematicae 251, (2020), 195-202. https://doi.org/10.4064/fm869-2-2020

C. Corral and M. Hrušák, Fréchet-like properties and almost disjoint families, Topology Appl. 277 (2020), 107216. https://doi.org/10.1016/j.topol.2020.107216

A. Dow, More set-theory for topologists, Topology Appl. 64, no. 3 (1995), 243-300. https://doi.org/10.1016/0166-8641(95)00034-E

G. Gruenhage, Products of Fréchet spaces, Topology Proc. 30, no. 2 (2006), 475-499.

G. Gruenhage, Infinite games and generalizations of first-countable spaces, General Topology Appl. 6, no. 3 (1976), 339-352. https://doi.org/10.1016/0016-660X(76)90024-6

G. Gruenhage and P. J. Szeptycki, Fréchet-Urysohn for finite sets, Topology Appl. 151, no. 1-3 (2005), 238-259. https://doi.org/10.1016/j.topol.2003.09.014

G. Gruenhage and P. J. Szeptycki, Fréchet-Urysohn for finite sets, II, Topology Appl. 154, no. 15 (2007), 2856-2872. https://doi.org/10.1016/j.topol.2005.11.021

F. Hernández-Hernández and M. Hrušák, Topology of Mrówka-Isbell spaces, in: M. Hrušák, M., Á. Tamariz-Mascarúa, M. Tkachenko (eds), Pseudocompact Topological Spaces. Developments in Mathematics, vol 55. Springer (2018). https://doi.org/10.1007/978-3-319-91680-4_8

M. Hrušák, Almost disjoint families and topology, in: K. Hart, J. van Mill, P. Simon (eds), Recent Progress in General Topology III. Atlantis Press, Paris (2014). https://doi.org/10.2991/978-94-6239-024-9_14

M. Hrušák and U. A. Ramos García, Malykhin's problem, Advances in Mathematics 262 (2014), 193-212. https://doi.org/10.1016/j.aim.2014.05.009

R. B. Jensen, The fine structure of the constructible hierarchy, Annals of Mathematical logic 4, no. 3 (1972), 229-308. https://doi.org/10.1016/0003-4843(72)90001-0

M. Magidor and C. Lambie-Hanson, On the strenhts and weaknesses of weak squares, Appalachian Set Theory 406 (2012), 301-330. https://doi.org/10.1017/CBO9781139208574.011

V. I. Malyhin, On countable spaces having no bicompactification of countable tightness, Soviet Math. Dokl. 13 (1972), 1407-1411.

E. A. Michael, A quintuple quotient quest, General Topology Appl. 2 (1972), 91-138. https://doi.org/10.1016/0016-660X(72)90040-2

S. Mrówka, On completely regular spaces, Fundamenta Mathematicae, 41.1, (1955), 105-106. https://doi.org/10.4064/fm-41-1-105-106

P. L. Sharma, Some characterizations of W-spaces and w-spaces, General Topology Appl. 9, no. 3 (1978), 289-293. https://doi.org/10.1016/0016-660X(78)90032-6

P. Simon, A compact Fréchet space whose square is not Fréchet, Commentationes Mathematicae Universitatis Carolinae 21, no. 4 (1980), 749-753.

O. V. Sipacheva, Spaces Fréchet-Urysohn with respect to families of subsets, Topology Appl. 121, no. 1-2 (2002), 305-317. https://doi.org/10.1016/S0166-8641(01)00125-0

S. Todorčević, Partitioning pairs of countable sets, Proc. Amer. Math. Soc. 111, no. 3 (1991), 841-844. https://doi.org/10.1090/S0002-9939-1991-1036992-4

S. Todorčević, A dichotomy for P-ideals of countable sets, Fundamenta Mathematicae 166, no. 3 (2000), 251-267. https://doi.org/10.4064/fm-166-3-251-267

S. Todorčević, Coherent sequences, in: M. Foreman, A. Kanamori (eds), Handbook of Set Theory. Springer, Dordrecht (2010).