Topological spaces compact with respect to a set of filters

• Autorzy: Paolo Lipparini
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Filter; Ultrafilter convergence; Sequencewise -compactness; Preservation under products; Comfort pre-order; Sequential compactness

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2014
• Tom: 12
• Numer: 7
• Strony: 991-999

Daty

wydano

2014-07-01

online

2014-04-03

Twórcy

autor

Paolo Lipparini

• II Università di Roma (Tor Vergata),

Bibliografia

• [1] Alexandroff P., Urysohn P., Mémorie sur les Espaces Topologiques Compacts, Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam, 14, Koninklijke Akademie van Wetenschappen, Amsterdam, 1929
• [2] Bernstein A.R., A new kind of compactness for topological spaces, Fund. Math., 1969/1970, 66, 185–193
• [3] Brandhorst S., Tychonoff-Like Theorems and Hypercompact Topological Spaces, Bachelor’s thesis, Leibniz Universität, Hannover, 2013
• [4] Caicedo X., The abstract compactness theorem revisited, In: Logic and Foundations of Mathematics, Florence, August, 1995, Synthese Lib., 280, Kluwer, Dordrecht, 1999, 131–141
• [5] Choquet G., Sur les notions de filtre et de grille, C. R. Acad. Sci. Paris, 1947, 224, 171–173
• [6] van Douwen E.K., The integers and topology, In: Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, 1984, 111–167
• [7] García-Ferreira S., Some remarks on initial α-compactness, < α-boundedness and p-compactness, Topology Proc., 1990, 15, 11–28
• [8] García-Ferreira S., On FU(p)-spaces and p-sequential spaces, Comment. Math. Univ. Carolin., 1991, 32(1), 161–171
• [9] García-Ferreira S., Quasi M-compact spaces, Czechoslovak Math. J., 1996, 46(121)(1), 161–177
• [10] García-Ferreira S., Kočinac Lj., Convergence with respect to ultrafilters: a survey, Filomat, 1996, 10, 1–32
• [11] Ginsburg J., Saks V., Some applications of ultrafilters in topology, Pacific J. Math., 1975, 57(2), 403–418 http://dx.doi.org/10.2140/pjm.1975.57.403
• [12] Katětov M., Products of filters, Comment. Math. Univ. Carolin., 1968, 9, 173–189
• [13] Kombarov A.P., Compactness and sequentiality with respect to a set of ultrafilters, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1985, 5, 15–18 (in Russian)
• [14] Lipparini P., More generalizations of pseudocompactness, Topology Appl., 2011, 158(13), 1655–1666 http://dx.doi.org/10.1016/j.topol.2011.05.039
• [15] Lipparini P., Some compactness properties related to pseudocompactness and ultrafilter convergence, Topol. Proc., 2012, 40, 29–51
• [16] Lipparini P., A very general covering property, Comment. Math. Univ. Carolin., 2012, 53(2), 281–306
• [17] Lipparini P., A characterization of the Menger property by means of ultrafilter convergence, Topology Appl., 2013, 160(18), 2505–2513 http://dx.doi.org/10.1016/j.topol.2013.07.044
• [18] Lipparini P., Productivity of [µ; λ]-compactness, preprint available at http://arxiv.org/abs/1210.2121
• [19] Saks V., Ultrafilter invariants in topological spaces, Trans. Amer. Math. Soc., 1978, 241, 79–97 http://dx.doi.org/10.1090/S0002-9947-1978-0492291-9
• [20] Smirnov Yu.M., On topological spaces compact in a given interval of powers, Izvestiya Akad. Nauk SSSR Ser. Mat., 1950, 14, 155–178 (in Russian)

Identyfikatory

DOI

10.2478/s11533-013-0398-2