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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 [[Cardinal number|Cardinal number]]), and consider the following statement:
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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:
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$G ( \kappa , \lambda )$: 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 $\xi &lt; \eta &lt; \kappa$;
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1) $U _ { \xi } \subset _{*} U _ { \eta }$ if $\xi < \eta < \kappa$;
  
2) $V _ { \xi } \subset_{*} V _ { \eta }$ if $\xi &lt; \eta &lt; \lambda$;
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2) $V _ { \xi } \subset_{*} V _ { \eta }$ if $\xi < \eta < \lambda$;
  
3) if $\xi &lt; \kappa$ and $\eta &lt; \lambda$, then $U _ { \xi } \cap V _ { \eta } =_{*}  \emptyset$;
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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 $\xi &lt; \kappa$ and $W \cap U _ { \xi } =_{*} \emptyset$ for all $\xi &lt; \lambda$.
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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.
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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>
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<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>
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<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>
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<tr><td valign="top">[a3]</td> <td valign="top">  K. Kunen,  "$( \kappa , \lambda ^ { * } )$-gaps under MA"  ''Unpublished manuscript''</td></tr>
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</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
How to Cite This Entry:
Hausdorff gap. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hausdorff_gap&oldid=50608
This article was adapted from an original article by J. van Mill (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article