Hausdorff gap
If 
and 
are subsets of 
, then one writes 
provided that 
is finite. In addition, 
means that 
while, moreover, 
is infinite. Finally, 
means that 
is finite.
Let 
and 
be infinite cardinal numbers (cf. also Cardinal number), and consider the following statement:
: There are a 
-sequence 
of subsets of 
and a 
-sequence 
of subsets of 
such that:
1) 
if 
;
2) 
if 
;
3) if 
and 
, then 
;
4) there does not exist a subset 
of 
such that 
for all 
and 
for all 
.
In [a2], F. Hausdorff proved that 
) is false while 
) is true. The sets that witness the fact that 
) 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 
) and 
) 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 
) and 
) 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  Mengen" Fund. Math. , 26 (1936) pp. 241–255 |
| [a3] | K. Kunen, " -gaps under MA" Unpublished manuscript |
Hausdorff gap. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hausdorff_gap&oldid=50381