129 research outputs found
On union ultrafilters
We present some new results on union ultrafilters. We characterize stability
for union ultrafilters and, as the main result, we construct a new kind of
unordered union ultrafilter
Quasi-selective ultrafilters and asymptotic numerosities
We isolate a new class of ultrafilters on N, called "quasi-selective" because
they are intermediate between selective ultrafilters and P-points. (Under the
Continuum Hypothesis these three classes are distinct.) The existence of
quasi-selective ultrafilters is equivalent to the existence of "asymptotic
numerosities" for all sets of tuples of natural numbers. Such numerosities are
hypernatural numbers that generalize finite cardinalities to countable point
sets. Most notably, they maintain the structure of ordered semiring, and, in a
precise sense, they allow for a natural extension of asymptotic density to all
sequences of tuples of natural numbers.Comment: 27 page
Homogeneous Subspaces of Products of Extremally Disconnected Spaces
Homogeneous countably compact spaces and whose product is
not pseudocompact are constructed. It is proved that all compact subsets of
homogeneous subspaces of the third power of an extremally disconnected space
are finite. Moreover, under CH, all compact subsets of homogeneous subspaces of
any finite power of an extremally disconnected space are finite and all compact
subsets of homogeneous subspaces of the countable power of an extremally
disconnected space are metrizable. It is also proved that all compact
homogeneous subspaces of finite powers of an extremally disconnected space are
finite, which strengthens Frol\'{\i}k's theorem
Quasi-selective ultrafilters and asymptotic numerosities
We isolate a new class of ultrafilters on N, called “quasi-selective” because they are intermediate between selective ultrafilters and P-points. (Under the Continuum Hypothesis these three classes are distinct.) The existence of quasi-selective ultrafilters is equivalent to the existence of “asymptotic numerosities” for all sets of tuples A ⊆ N^k. Such numerosities are hypernatural numbers that generalize finite cardinalities to countable point sets. Most notably, they maintain the structure of ordered semiring, and, in a precise sense, they allow for a natural extension of asymptotic density to all sets of tuples of natural numbers
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