129 research outputs found

    On union ultrafilters

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    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

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    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

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    Homogeneous countably compact spaces XX and YY whose product XĂ—YX\times Y 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

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    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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