Difference between revisions of "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 $\{ U _ { \xi } : \xi < \kappa \}$ 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 $\xi < \eta < \lambda$; | |
| − | In [[#References|[a2]]], F. Hausdorff proved that | + | 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 [[#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==== | ====References==== | ||
| − | <table>< | + | <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> |
Revision as of 17:00, 1 July 2020
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 |
How to Cite This Entry:
Hausdorff gap. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hausdorff_gap&oldid=13106
Hausdorff gap. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hausdorff_gap&oldid=13106
This article was adapted from an original article by J. van Mill (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article