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Hausdorff ultrafilters
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We give the name Hausdorff to those ultrafilters that provide ultrapowers whose natural topology (
S
S
-topology) is Hausdorff, e.g. selective ultrafilters are Hausdorff. Here we give necessary and sufficient conditions for product ultrafilters to be Hausdorff. Moreover we show that no regular ultrafilter over the “small” uncountable cardinal
u
\mathfrak {u}
can be Hausdorff. (
u
\mathfrak {u}
is the least size of an ultrafilter basis on
ω
\omega
.) We focus on countably incomplete ultrafilters, but our main results also hold for
κ
\kappa
-complete ultrafilters.
American Mathematical Society (AMS)
Title: Hausdorff ultrafilters
Description:
We give the name Hausdorff to those ultrafilters that provide ultrapowers whose natural topology (
S
S
-topology) is Hausdorff, e.
g.
selective ultrafilters are Hausdorff.
Here we give necessary and sufficient conditions for product ultrafilters to be Hausdorff.
Moreover we show that no regular ultrafilter over the “small” uncountable cardinal
u
\mathfrak {u}
can be Hausdorff.
(
u
\mathfrak {u}
is the least size of an ultrafilter basis on
ω
\omega
.
) We focus on countably incomplete ultrafilters, but our main results also hold for
κ
\kappa
-complete ultrafilters.
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