369 research outputs found
Countable dense homogeneous filters and the Menger covering property
In this note we present a ZFC construction of a non-meager filter which fails
to be countable dense homogeneous. This answers a question of
Hern\'andez-Guti\'errez and Hru\v{s}\'ak. The method of the proof also allows
us to obtain a metrizable Baire topological group which is strongly locally
homogeneous but not countable dense homogeneous.Comment: 8 page
Menger's covering property and groupwise density
We establish a surprising connection between Menger's classical covering
property and Blass's modern combinatorial notion of groupwise density. This
connection implies a short proof of the groupwise density bound on the
additivity number for Menger's property.Comment: Small update
On critical cardinalities related to -sets
In this note we collect some known information and prove new results about
the small uncountable cardinal . The cardinal is
defined as the smallest cardinality of a subset
which is not a -set (a subspace is called a -set if
each subset is of type in ). We present a simple
proof of a folklore fact that , and also establish the
consistency of a number of strict inequalities between the cardinal and other standard small uncountable cardinals. This is done by combining
some known forcing results. A new result of the paper is the consistency of
, where denotes
the linear refinement number. Another new result is the upper bound holding for any -flexible cccc
-ideal on .Comment: 8 page
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