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COMPLETE CCC BOOLEAN ALGEBRAS, THE ORDER SEQUENTIAL TOPOLOGY, AND A PROBLEM OF VON NEUMANN

Published online by Cambridge University Press:  12 December 2005

B. BALCAR ,
T. JECH  and
T. PAZÁK
B. BALCAR
Affiliation:
Žitná 25, 115 67 Praha 1, Czech Republicbalcar@math.cas.cz, jech@math.cas.cz, pazak@math.cas.cz
T. JECH
Affiliation:
Žitná 25, 115 67 Praha 1, Czech Republicbalcar@math.cas.cz, jech@math.cas.cz, pazak@math.cas.cz
T. PAZÁK
Affiliation:
Žitná 25, 115 67 Praha 1, Czech Republicbalcar@math.cas.cz, jech@math.cas.cz, pazak@math.cas.cz
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Abstract

Let B be a complete ccc Boolean algebra and let $\tau_s$ be the topology on B induced by the algebraic convergence of sequences in B.

1. Either there exists a Maharam submeasure on B or every nonempty open set in $(B,\tau_s)$ is topologically dense.

2. It is consistent that every weakly distributive complete ccc Boolean algebra carries a strictly positive Maharam submeasure.

3. The topological space $(B,\tau_s)$ is sequentially compact if and only if the generic extension by B does not add independent reals.

Examples are also given of ccc forcings adding a real but not independent reals.

Keywords

28A6006E10 (primary)03E5554A2054A25 (secondry)
Type
Papers
Information
Bulletin of the London Mathematical Society , Volume 37 , Issue 6 , December 2005 , pp. 885 - 898
Copyright
© The London Mathematical Society 2005

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Footnotes

Supported in part by the GAČR Grant number 201/02/0857 (Balcar and Jech) and the GAČR Grant number 201/03/0933 (Pazák).