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Keywords:
complete Boolean algebra; convergence structure; algebraic convergence; forcing; Cantor cube; Aleksandrov cube; small cardinal
Summary:
We compare the forcing-related properties of a complete Boolean algebra ${\mathbb B}$ with the properties of the convergences $\lambda _{\mathrm s}$ (the algebraic convergence) and $\lambda _{\mathrm {ls}}$ on ${\mathbb B}$ generalizing the convergence on the Cantor and Aleksandrov cube, respectively. In particular, we show that $\lambda _{\mathrm {ls}}$ is a topological convergence iff forcing by ${\mathbb B}$ does not produce new reals and that $\lambda _{\mathrm {ls}}$ is weakly topological if ${\mathbb B}$ satisfies condition $(\hbar )$ (implied by the ${\mathfrak t}$-cc). On the other hand, if $\lambda _{\mathrm {ls}}$ is a weakly topological convergence, then ${\mathbb B}$ is a $2^{\mathfrak h}$-cc algebra or in some generic extension the distributivity number of the ground model is greater than or equal to the tower number of the extension. So, the statement “The convergence $\lambda _{\mathrm {ls}}$ on the collapsing algebra ${\mathbb B}=\mathop {\mathrm {ro}} (^{<\omega }\omega _2)$ is weakly topological“ is independent of ZFC.
References:
[1] Balcar, B., Główczyński, W., Jech, T.: The sequential topology on complete Boolean algebras. Fundam. Math. 155 (1998), 59-78. MR 1487988 | Zbl 0910.28004
[2] Balcar, B., Jech, T., Pazák, T.: Complete ccc Boolean algebras, the order sequential topology, and a problem of von Neumann. Bull. Lond. Math. Soc. 37 (2005), 885-898. DOI 10.1112/S0024609305004807 | MR 2186722 | Zbl 1101.28003
[3] Balcar, B., Jech, T.: Weak distributivity, a problem of von Neumann and the mystery of measurability. Bull. Symb. Log. 12 (2006), 241-266. DOI 10.2178/bsl/1146620061 | MR 2223923 | Zbl 1120.03028
[4] Balcar, B., Pelant, J., Simon, P.: The space of ultrafilters on $N$ covered by nowhere dense sets. Fundam. Math. 110 (1980), 11-24. MR 0600576 | Zbl 0568.54004
[5] Engelking, R.: General Topology. Translated from the Polish. Sigma Series in Pure Mathematics 6 Heldermann, Berlin (1989). MR 1039321
[6] Farah, I.: Examples of $\varepsilon$-exhaustive pathological submeasures. Fundam. Math. 181 (2004), 257-272. DOI 10.4064/fm181-3-4 | MR 2099603 | Zbl 1069.28002
[7] Jech, T.: Set Theory. Perspectives in Mathematical Logic Springer, Berlin (1997). MR 1492987 | Zbl 0882.03045
[8] Kunen, K.: Set Theory. An Introduction to Independence Proofs. Studies in Logic and the Foundations of Mathematics 102 North-Holland, Amsterdam (1980). MR 0597342 | Zbl 0443.03021
[9] Kurilić, M. S., Pavlović, A.: A posteriori convergence in complete Boolean algebras with the sequential topology. Ann. Pure Appl. Logic 148 (2007), 49-62. DOI 10.1016/j.apal.2007.05.002 | MR 2352578 | Zbl 1132.06008
[10] Kurilić, M. S., Pavlović, A.: Some forcing related convergence structures on complete Boolean algebras. Novi Sad J. Math. 40 (2010), 77-94. MR 2827660 | Zbl 1265.54131
[11] Kurilić, M. S., Pavlović, A.: The convergence of the sequences coding the ground model reals. Publ. Math. Debrecen 82 (2013), 277-292. DOI 10.5486/PMD.2013.5199 | MR 3034346
[12] Kurilić, M. S., Todorčević, S.: Property $(\hbar)$ and cellularity of complete Boolean algebras. Arch. Math. Logic 48 (2009), 705-718. DOI 10.1007/s00153-009-0144-4 | MR 2563812 | Zbl 1201.03044
[13] Maharam, D.: An algebraic characterization of measure algebras. Ann. Math. (2) 48 (1947), 154-167. DOI 10.2307/1969222 | MR 0018718 | Zbl 0029.20401
[14] (ed.), R. D. Mauldin: The Scottish Book. Mathematics from the Scottish Café. Birkhäuser, Boston (1981). MR 0666400 | Zbl 0485.01013
[15] Talagrand, M.: Maharam's problem. C. R., Math., Acad. Sci. Paris 342 (2006), 501-503. DOI 10.1016/j.crma.2006.01.026 | MR 2214604 | Zbl 1099.28004
[16] Talagrand, M.: Maharam's problem. Ann. Math. (2) 168 (2008), 981-1009. MR 2456888 | Zbl 1185.28002
[17] Todorcevic, S.: A problem of von Neumann and Maharam about algebras supporting continuous submeasures. Fundam. Math. 183 (2004), 169-183. DOI 10.4064/fm183-2-7 | MR 2127965 | Zbl 1071.28004
[18] Douwen, E. K. van: The integers and topology. Handbook of Set-Theoretic Topology K. Kunen, J. E. Vaughan North-Holland, Amsterdam (1984), 111-167. MR 0776622
[19] Velickovic, B.: ccc forcing and splitting reals. Isr. J. Math. 147 (2005), 209-220. DOI 10.1007/BF02785365 | MR 2166361 | Zbl 1118.03046
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