Ultraproducts, \(p\)-limits and antichains on the Comfort group order. (English) Zbl 1071.54004
The authors of the paper under review consider the following question: Does \({\mathbf M}{\mathbf A}\) imply that the Comfort group order is not downward directed? The answer “yes” is equivalent to the existence of a \(p\)-compact group \(G\) and a \(q\)-compact group \(H\) for which \(G\times H\) is not \(r\)-compact for any free ultrafilter \(r\) on \(\omega\).
The authors construct \(p\)-compact groups as follows: Let \(p\), \(q\) be incomparable selective ultrafilters. Then there exists a \(p\)-compact group \(G\) and a \(q\)-compact group \(H\) for which \(G\times H\) is not countably compact. This implies the following important results: if there are \(2^c\) incomparable selective ultrafilters, then the Comfort group order has an antichain of size \(2^c\). \({\mathbf M}{\mathbf A}\) implies the Comfort group order has an antichain of size \(2^c\).
The authors construct \(p\)-compact groups as follows: Let \(p\), \(q\) be incomparable selective ultrafilters. Then there exists a \(p\)-compact group \(G\) and a \(q\)-compact group \(H\) for which \(G\times H\) is not countably compact. This implies the following important results: if there are \(2^c\) incomparable selective ultrafilters, then the Comfort group order has an antichain of size \(2^c\). \({\mathbf M}{\mathbf A}\) implies the Comfort group order has an antichain of size \(2^c\).
Reviewer: Kiyoshi Iseki (Osaka)
MSC:
| 54B10 | Product spaces in general topology |
| 03E50 | Continuum hypothesis and Martin’s axiom |
| 54A35 | Consistency and independence results in general topology |
| 54G20 | Counterexamples in general topology |
Keywords:
\(p\)-limit; \(p\)-compact; Comfort group order; Countably compact group; Convergence; Topological product; Selective ultrafiltersReferences:
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