On 𝜎-countably tight spaces

Juhász, István; van Mill, Jan

doi:10.1090/proc/13682

On $\sigma$-countably tight spaces
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by István Juhász and Jan van Mill PDF

Proc. Amer. Math. Soc. 146 (2018), 429-437 Request permission

Abstract:

Extending a result of R. de la Vega, we prove that an infinite homogeneous compactum has cardinality $\mathfrak {c}$ if it is the union of either countably many dense or finitely many arbitrary countably tight subspaces. The question if every infinite homogeneous and $\sigma$-countably tight compactum has cardinality $\mathfrak {c}$ remains open.

We also show that if an arbitrary product is $\sigma$-countably tight, then all but finitely many of its factors must be countably tight.

References

A. V. Arhangel’skii and J. van Mill, Topological homogeneity, Recent progress in general topology. III, Atlantis Press, Paris, 2014, pp. 1–68. MR 3204728, DOI 10.2991/978-94-6239-024-9_{1}
Raushan Z. Buzyakova, Cardinalities of some Lindelöf and $\omega _1$-Lindelöf $T_1/T_2$-spaces, Topology Appl. 143 (2004), no. 1-3, 209–216. MR 2080291, DOI 10.1016/j.topol.2004.02.017
N. A. Carlson and G. J. Ridderbos, On several cardinality bounds on power homogeneous spaces, Houston J. Math. 38 (2012), no. 1, 319–332. MR 2917289, DOI 10.1007/s11118-012-9278-9
Ramiro de la Vega, A new bound on the cardinality of homogeneous compacta, Topology Appl. 153 (2006), no. 12, 2118–2123. MR 2239075, DOI 10.1016/j.topol.2005.08.005
Ryszard Engelking, General topology, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from the Polish by the author. MR 1039321
István Juhász, Cardinal functions in topology—ten years later, 2nd ed., Mathematical Centre Tracts, vol. 123, Mathematisch Centrum, Amsterdam, 1980. MR 576927
Jan van Mill, On the cardinality of power homogeneous compacta, Topology Appl. 146/147 (2005), 421–428. MR 2107161, DOI 10.1016/j.topol.2003.07.020