Some applications of a generalized Martin's axiom
This paper is dedicated to the memory of B. Shapirovskij. The first half of this paper is mainly concerned with proposition \(P_ 1\) which is implied by Baumgartner axiom plus CH. By \(P_ 1\) we mean the following statement: Suppose \(\{A_ \alpha\}_{\alpha < k}\), \(k < 2^{\aleph_ 1}\), are subsets of \(\omega_ 1\) with each countable intersection uncountable. Then there is an uncountable \(A \subset \omega_ 1\) such that for every \(\alpha\), \(A - A_ \alpha\) is countable. A Lusin set style formulation of \(P_ 1\) is given as useful. Under \(P_ 1\) plus CH plus for each \(k < 2^{\aleph_ k}\), \(k^{\aleph_ 0} < 2^{\aleph_ 1}\), the generalized Martin's axiom holds for countably compact \(\aleph_ 1\)-centred partial orders. Under \(P_ 1\) and CH, there exists a hereditarily Lindelöf regular nonseparable space of weight \(k\) for which \(\aleph_ 2 \leq k^{\aleph_ 0} < 2^{\aleph_ 1}\). Under \(P_ 1\) plus CH plus \(2^{\aleph_ 1} > k \geq \aleph_ 1\), we have \(2^ K = 2^{\aleph_ 1}\). The second half is concerned with the strong Baumgartner axiom. Some results under the strong Baumgartner axiom plus CH are also proved. The author gives a model for CH and the generalized Martin's axiom for \(\aleph_ 1\)-centred countably compact partial orders plus \(2^{\aleph_ 1} > \aleph_ 2\) in which any normal space of character \(< 2^{\aleph_ 1}\) is collectionwise normal with respect to a discrete collection of \(\aleph_ 1\) Lindelöf sets, each of cardinality \(< 2^{\aleph_ 1}\). To state exactly the obtained results, we need several concepts, so the details are not mentioned here. On a Kurepa tree, there are two interesting results. Under the Baumgartner axiom, CH and \(2^{\aleph_ 1} > \aleph_ 2\) implies there is a Kurepa tree. On the other hand, \(P_ 1\) and CH implies that there is a Kurepa tree. This paper is written in the form of an expository/survey article style.
scientific article; zbMATH DE number 600039
scientific article