A solution to the L space problem

Moore, Justin ¹

¹ Department of Mathematics, Boise State University, Boise, Idaho 83725

Journal of the American Mathematical Society, Tome 19 (2006) no. 3, pp. 717-736

Résumé

In this paper I will construct a non-separable hereditarily Lindelöf space (L space) without any additional axiomatic assumptions. The constructed space $\mathscr {L}$ is a subspace of ${\mathbb {T}}^{\omega _1}$ where $\mathbb {T}$ is the unit circle. It is shown to have a number of properties which may be of additional interest. For instance it is shown that the closure in $\mathbb {T}^{\omega _1}$ of any uncountable subset of $\mathscr {L}$ contains a canonical copy of $\mathbb {T}^{\omega _1}$. I will also show that there is a function $f:[\omega _1]^2 \to \omega _1$ such that if $A,B \subseteq \omega _1$ are uncountable and $\xi \omega _1$, then there are $\alpha \beta$ in $A$ and $B$ respectively with $f (\alpha ,\beta ) = \xi$. Previously it was unknown whether such a function existed even if $\omega _1$ was replaced by $2$. Finally, I will prove that there is no basis for the uncountable regular Hausdorff spaces of cardinality $\aleph _1$. The results all stem from the analysis of oscillations of coherent sequences $\langle e_\beta :\beta \omega _1\rangle$ of finite-to-one functions. I expect that the methods presented will have other applications as well.

Détail

DOI : 10.1090/S0894-0347-05-00517-5

Bibliographie

[1] Cassels, J. W. S. An introduction to Diophantine approximation 1957

[2] Fremlin, D. H. Consequences of Martin’s axiom 1984

[3] Gruenhage, Gary Perfectly normal compacta, cosmic spaces, and some partition problems 1990 85 95

[4] Hajnal, A., Juhász, I. On hereditarily 𝛼-Lindelöf and hereditarily 𝛼-separable spaces Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 1968 115 124

[5] Jech, T. Multiple forcing 1986

[6] Juhász, I. A survey of 𝑆- and 𝐿-spaces 1980 675 688

[7] Kunen, Kenneth Strong 𝑆 and 𝐿 spaces under 𝑀𝐴 1977 265 268

[8] Kunen, Kenneth Set theory 1983

[9] Roitman, Judy Basic 𝑆 and 𝐿 1984 295 326

[10] Rudin, Mary Ellen 𝑆 and 𝐿 spaces 1980 431 444

[11] Szentmiklóssy, Z. 𝑆-spaces and 𝐿-spaces under Martin’s axiom 1980 1139 1145

[12] Todorčević, Stevo Forcing positive partition relations Trans. Amer. Math. Soc. 1983 703 720

[13] Todorčević, Stevo Partitioning pairs of countable ordinals Acta Math. 1987 261 294

[14] Todorčević, Stevo Oscillations of real numbers 1988 325 331

[15] Todorčević, Stevo Partition problems in topology 1989

[16] Todorcevic, Stevo A classification of transitive relations on 𝜔₁ Proc. London Math. Soc. (3) 1996 501 533

[17] Todorcevic, Stevo Basis problems in combinatorial set theory Doc. Math. 1998 43 52

[18] Tukey, John W. Convergence and Uniformity in Topology 1940

[19] Vojtáš, Peter Generalized Galois-Tukey-connections between explicit relations on classical objects of real analysis 1993 619 643

[20] Zenor, Phillip Hereditary 𝔪-separability and the hereditary 𝔪-Lindelöf property in product spaces and function spaces Fund. Math. 1980 175 180