View source for Hausdorff gap

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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|Cardinal number]]), and consider the following statement:

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

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

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

3) if $\xi &lt; \kappa$ and $\eta &lt; \lambda$, 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 $\xi &lt; \kappa$ and $W \cap U _ { \xi } =_{*} \emptyset$ for all $\xi &lt; \lambda$.

In [[#References|[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 [[#References|[a3]]] that it is consistent with Martin's axiom (cf. also [[Suslin hypothesis|Suslin hypothesis]]) and the negation of the [[Continuum hypothesis|continuum hypothesis]] that $G ( \omega _ { 1 } , c )$) and $G ( \mathfrak c , \mathfrak c )$) both are false. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120040/h12004038.png"/> is the cardinality of the continuum (cf. also [[Continuum, cardinality of the|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 [[#References|[a1]]] for more details.

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  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</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  F. Hausdorff,   "Summen von $\aleph_1$ Mengen"  ''Fund. Math.'' , '''26'''  (1936)  pp. 241–255</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  K. Kunen,   "$( \kappa , \lambda ^ { * } )$-gaps under MA"  ''Unpublished manuscript''</td></tr></table>