Two applications of reflection and forcing to topology. (English) Zbl 0641.54002
General topology and its relations to modern analysis and algebra VI, Proc. 6th Symp., Prague/Czech. 1986, Res. Expo. Math. 16, 155-172 (1988).
[For the entire collection see Zbl 0632.00016.]
Extending the use of reflection arguments to situations with no or minimal large cardinal assumptions, the following statements are shown consistent with not-CH: (A) If X is comact and all \(Y\in [X]^{\omega_ 1}\) are metrizable, then X is metrizable. \((Here[X]^{\omega_ 1}=\{Y\subset X: | Y| =\omega_ 1\}.)\) (B) Every first countable space with a small diagonal is metrizable. (Here X has small diagonal \(\Delta\) iff for all \(Y\in [X-\Delta]^{\omega_ 1}\) there is an open \(u\supset \Delta\) with \(| Y-u| =\omega_ 1.)\) The consistency of (B) \(+\) not-CH is relative to a weakly compact cardinal.
The author is as concerned with presenting the method (reflection-with- forcing) as the theorems, and the paper is therefore of extremely broad interest in the field of set-theoretic topology.
Extending the use of reflection arguments to situations with no or minimal large cardinal assumptions, the following statements are shown consistent with not-CH: (A) If X is comact and all \(Y\in [X]^{\omega_ 1}\) are metrizable, then X is metrizable. \((Here[X]^{\omega_ 1}=\{Y\subset X: | Y| =\omega_ 1\}.)\) (B) Every first countable space with a small diagonal is metrizable. (Here X has small diagonal \(\Delta\) iff for all \(Y\in [X-\Delta]^{\omega_ 1}\) there is an open \(u\supset \Delta\) with \(| Y-u| =\omega_ 1.)\) The consistency of (B) \(+\) not-CH is relative to a weakly compact cardinal.
The author is as concerned with presenting the method (reflection-with- forcing) as the theorems, and the paper is therefore of extremely broad interest in the field of set-theoretic topology.
Reviewer: J.Roitman
MSC:
| 54A25 | Cardinality properties (cardinal functions and inequalities, discrete subsets) |
| 54E35 | Metric spaces, metrizability |
| 03E75 | Applications of set theory |
| 54A35 | Consistency and independence results in general topology |
