530 research outputs found
Seven characterizations of non-meager P-filters
We give several topological/combinatorial conditions that, for a filter on
, are equivalent to being a non-meager -filter. In
particular, we show that a filter is countable dense homogeneous if and only if
it is a non-meager -filter. Here, we identify a filter with a
subspace of through characteristic functions. Along the way, we
generalize to non-meager -filters a result of Miller about
-points, and we employ and give a new proof of results of
Marciszewski. We also employ a theorem of Hern\'andez-Guti\'errez and
Hru\v{s}\'ak, and answer two questions that they posed. Our result also
resolves several issues raised by Medini and Milovich, and proves false one
"theorem" of theirs. Furthermore, we show that the statement "Every non-meager
filter contains a non-meager -subfilter" is independent of
(more precisely, it is a consequence of
and its negation is a consequence of ).
It follows from results of Hru\v{s}\'ak and van Mill that, under
, a filter has less than types of
countable dense subsets if and only if it is a non-meager -filter.
In particular, under , there exists an ultrafilter
with types of countable dense subsets. We also show that such an
ultrafilter exists under .Comment: 17 page
On well-splitting posets
We introduce a class of proper posets which is preserved under countable
support iterations, includes -bounding, Cohen, Miller, and
Mathias posets associated to filters with the Hurewicz covering properties, and
has the property that the ground model reals remain splitting and unbounded in
corresponding extensions. Our results may be considered as a possible path
towards solving variations of the famous Roitman problem.Comment: 10 pages. This is the submitted version, but it is rather close to
the accepted on
On the lattice of weak topologies on the bicyclic monoid with adjoined zero
A Hausdorff topology on the bicyclic monoid with adjoined zero
is called {\em weak} if it is contained in the coarsest inverse
semigroup topology on . We show that the lattice
of all weak shift-continuous topologies on is isomorphic to the
lattice of all shift-invariant filters on with an attached element
endowed with the following partial order: iff
or . Also, we investigate
cardinal characteristics of the lattice . In particular, we proved
that contains an antichain of cardinality and
a well-ordered chain of cardinality . Moreover, there exists a
well-ordered chain of first-countable weak topologies of order type
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