A note on rank 2 diagonals
Keywords:cardinality bounds, weakly Lindelöf, Gδ-diagonal, neighbourhood assignment, dual properties
We solve two questions regarding spaces with a (Gδ)-diagonal of rank 2. One is a question of Basile, Bella and Ridderbos about weakly Lindelöf spaces with a Gδ-diagonal of rank 2 and the other is a question of Arhangel’skii and Bella asking whether every space with a diagonal of rank 2 and cellularity continuum has cardinality at most continuum.
A. V. Arhangel'skii and A. Bella, The diagonal of a first-countable paratopological groups, submetrizability and related results, Appl. Gen. Topol. 8 (2007), 207-212. https://doi.org/10.4995/agt.2007.1881
A. V. Arhangel'skii and R. Z. Buzyakova, The rank of the diagonal and submetrizability, Comment. Math. Univ. Carolinae 47 (2006), 585-597.
A. Bella, Remarks on the metrizability degree, Boll. Union. Mat. Ital. 1-3 (1987), 391-396.
D. Basile, A. Bella and G. J. Ridderbos, Weak extent, submetrizability and diagonal degrees, Houston J. Math. 40 (2014), 255-266.
M. Bell, J. Ginsburg and G. Woods, Cardinal inequalities for topological spaces involving the weak Lindelöf number, Pacific J. Math. 79 (1978), no. 1, 37-45. https://doi.org/10.2140/pjm.1978.79.37
A. Bella and S. Spadaro, Cardinal invariants of cellular Lindelöf spaces, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113 (2019), 2805-2811. https://doi.org/10.1007/s13398-019-00660-1
R. Buzyakova, Cardinalities of ccc spaces with regular Gδ-diagonals, Topology Appl. 153 (2006), 1696-1698. https://doi.org/10.1016/j.topol.2005.06.004
J. Chaber, Conditions which imply compactness in countably compact spaces, Bull. Acad. Pol. Sci. Ser. Math. 24 (1976), 993-998.
E. K. van Douwen and M. Reed, On chain conditions in Moore spaces II, Topology Appl. 39 (1991), 65-69. https://doi.org/10.1016/0166-8641(91)90076-X
J. Ginsburg and R. G. Woods, A cardinal inequality for topological spaces involving closed discrete sets, Proc. Amer. Math. Soc. 64 (1977), 357-360. https://doi.org/10.1090/S0002-9939-1977-0461407-7
J. van Mill, V. V. Tkachuk and R. G. Wilson, Classes defined by stars and neighbourhood assignments, Topology Appl. 154, no. 10 (2007), 2127-2134. https://doi.org/10.1016/j.topol.2006.03.029
R. Engelking, General Topology, Heldermann Verlag, Berlin, second ed., 1989.
D. Shakhmatov, No upper bound for cardinalities of Tychonoff C.C.C. spaces with a Gδ diagonal exist (an answer to J. Ginsburg and R.G. Woods' question), Comment. Math. Univ. Carolinae 25 (1984), 731-746.
V. Sneider, Continuous images of Souslin and Borel sets: metrization theorems, Dokl. Acad. Nauk USSR, 50 (1945), 77-79.
V. Uspenskij, A large Fσ-discrete Fréchet space having the Souslin property, Comment. Math. Univ. Carolinae 25 (1984), 257-260.
W.-F. Xuan and Y.-K. Song, Dually properties and cardinal inequalities, Topology Appl. 234 (2018), 1-6. https://doi.org/10.1016/j.topol.2017.11.002
P. Zenor, On spaces with regular Gδ-diagonals, Pacific J. Math. 40 (1972), 959-963. https://doi.org/10.2140/pjm.1972.40.759
How to Cite
This journal is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.