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. | 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 [[ | + | 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 | + | $G ( \kappa , \lambda )$: 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 | + | 1) $U _ { \xi } \subset _{*} U _ { \eta }$ if $\xi < \eta < \kappa$; |
| − | 2) $V _ { \xi } \subset_{*} V _ { \eta }$ if $\xi | + | 2) $V _ { \xi } \subset_{*} V _ { \eta }$ if $\xi < \eta < \lambda$; |
| − | 3) if $\xi | + | 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 | + | 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 $G ( \omega , \omega )$ | + | 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]]) 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 [[#References|[a1]]] for more details. |
====References==== | ====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> | + | <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> | ||
Latest revision as of 20:23, 5 December 2023
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 |
Hausdorff gap. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hausdorff_gap&oldid=50381