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Compactness of Powers of ω

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Lipparini P.

Bulletin of the Polish Academy of Sciences. Mathematics

2013

tom 61

nr 3

239-246

We characterize exactly the compactness properties of the product of κ copies of the space ω with the discrete topology. The characterization involves uniform ultrafilters, infinitary languages, and the existence of nonstandard elements in elementary extensions. We also have results involving products of possibly uncountable regular cardinals.

Topological spaces compact with respect to a set of filters

100%

Lipparini P.

Open Mathematics

2014

tom 12

nr 7

991-999

If is a family of filters over some set I, a topological space X is sequencewise -compact if for every I-indexed sequence of elements of X there is such that the sequence has an F-limit point. Countable compactness, sequential compactness, initial κ-compactness, [λ; µ]-compactness, the Menger and Rothberger properties can all be expressed in terms of sequencewise -compactness for appropriate choices of . We show that sequencewise -compactness is preserved under taking products if and only if there is a filter such that sequencewise -compactness is equivalent to F-compactness. If this is the case, and there exists a sequencewise -compact T 1 topological space with more than one point, then F is necessarily an ultrafilter. The particular case of sequential compactness is analyzed in detail.

Ograniczanie wyników

1 Bulletin of the Polish Academy of Sciences. Mathematics

1 Open Mathematics

2 Lipparini P.

1 2014

1 2013

Wyniki wyszukiwania

Compactness of Powers of ω

Topological spaces compact with respect to a set of filters