Hausdorff gap

If and are subsets of , then one writes provided that is finite. In addition, means that while, moreover, is infinite. Finally, means that is finite.

Let and be infinite cardinal numbers (cf. also Cardinal number), and consider the following statement:

: There are a -sequence of subsets of and a -sequence of subsets of such that:

1) if ;

2) if ;

3) if and , then ;

4) there does not exist a subset of such that for all and for all .

In [a2], F. Hausdorff proved that ) is false while ) is true. The sets that witness the fact that ) holds are called a Hausdorff gap. K. Kunen has shown in [a3] that it is consistent with Martin's axiom (cf. also Suslin hypothesis) and the negation of the continuum hypothesis that ) and ) both are false. Here, is the cardinality of the continuum (cf. also Continuum, cardinality of the). He also proved that it is consistent with Martin's axiom and the negation of the continuum hypothesis that ) and ) both are true. See [a1] for more details.

References

[a1]	J.E. Baumgartner, "Applications of the Proper Forcing Axiom" K. Kunen (ed.) J.E. Vaughan (ed.) , Handbook of Set Theoretic Topology , North-Holland (1984) pp. 913–959
[a2]	F. Hausdorff, "Summen von Mengen" Fund. Math. , 26 (1936) pp. 241–255
[a3]	K. Kunen, "-gaps under MA" Unpublished manuscript