Hausdorff gap

If $A$ and $B$ are subsets of $\omega$, then one writes $A \subseteq_{*} B$ provided that $A \backslash B$ is finite. In addition, $A \subset_{*} B$ means that $A \subseteq_{*} B$ while, moreover, $B \backslash A$ is infinite. Finally, $A \cap B =_{*} \emptyset$ means that $A \cap B$ is finite.

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

$G ( \kappa , \lambda )$: There are a $\kappa$-sequence $\{ U _ { \xi } : \xi < \kappa \}$ of subsets of $\omega$ and a $\lambda$-sequence $\{ V _ { \xi } : \xi < \lambda \}$ of subsets of $\omega$ such that:

1) $U _ { \xi } \subset _{*} U _ { \eta }$ if $\xi < \eta < \kappa$;

2) $V _ { \xi } \subset_{*} V _ { \eta }$ if $\xi < \eta < \lambda$;

3) if $\xi < \kappa$ and $\eta < \lambda$, then $U _ { \xi } \cap V _ { \eta } =_{*} \emptyset$;

4) there does not exist a subset $W$ of $\omega$ such that $V _ { \xi } \subseteq_{ * } W$ for all $\xi < \kappa$ and $W \cap U _ { \xi } =_{*} \emptyset$ for all $\xi < \lambda$.

In [a2], F. Hausdorff proved that $G ( \omega , \omega )$) is false while $G ( \omega _ { 1 } , \omega _ { 1 } )$) is true. The sets that witness the fact that $G ( \omega _ { 1 } , \omega _ { 1 } )$) 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 $G ( \omega _ { 1 } , c )$) and $G ( \mathfrak c , \mathfrak c )$) 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 $G ( \omega _ { 1 } , c )$) and $G ( \mathfrak c , \mathfrak c )$) 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 $\aleph_1$ Mengen" Fund. Math. , 26 (1936) pp. 241–255
[a3]	K. Kunen, "$( \kappa , \lambda ^ { * } )$-gaps under MA" Unpublished manuscript