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, $\mathfrak c$ 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[a3]	K. Kunen, "$( \kappa , \lambda ^ { * } )$-gaps under MA" Unpublished manuscript