# Article

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Keywords:
covering property; subcover; compactness; accumulation point; convergence; pseudocompactness; limit point
Summary:
We introduce a general notion of covering property, of which many classical definitions are particular instances. Notions of closure under various sorts of convergence, or, more generally, under taking kinds of accumulation points, are shown to be equivalent to a covering property in the sense considered here (Corollary 3.10). Conversely, every covering property is equivalent to the existence of appropriate kinds of accumulation points for arbitrary sequences on some fixed index set (Corollary 3.5). We discuss corresponding notions related to sequential compactness, and to pseudocompactness, or, more generally, properties connected with the existence of limit points of sequences of subsets. In spite of the great generality of our treatment, many results here appear to be new even in very special cases, such as $D$-compactness and $D$-pseudocompactness, for $D$ an ultrafilter, and weak (quasi) $M$-(pseudo)-compactness, for $M$ a set of ultrafilters, as well as for $[\beta ,\alpha ]$-compactness, with $\beta$ and $\alpha$ ordinals.
References:
[1] Alexandroff P., Urysohn P.: Mémorie sur les éspaces topologiques compacts. Verh. Akad. Wetensch. Amsterdam 14 (1929), 1–96.
[2] Arens R., Dugundji J.: Remark on the concept of compactness. Portugaliae Math. 9 (1950), 141–143. MR 0038642 | Zbl 0039.18602
[3] Artico G., Marconi U., Pelant J., Rotter L., Tkachenko M.: Selections and suborderability. Fund. Math. 175 (2002), 1–33. MR 1971236 | Zbl 1019.54014
[4] Caicedo X.: The abstract compactness theorem revisited. in Logic and Foundations of Mathematics (A. Cantini et al. editors), Kluwer Academic Publishers, Dordrecht, 1999, pp. 131–141. MR 1739865 | Zbl 0955.03044
[5] Choquet C.: Sur les notions de filtre et de grille. C.R. Acad. Sci., Paris 224 (1947), 171–173. MR 0018813 | Zbl 0029.07602
[6] Dow A., Porter J.R., Stephenson R.M., Woods R.G.: Spaces whose pseudocompact subspaces are closed subsets. Appl. Gen. Topol. 5 (2004), 243–264. MR 2121792 | Zbl 1066.54024
[7] Engelking R.: General Topology. 2nd edition, Sigma Ser. Pure Math., 6, Heldermann, Berlin, 1989. MR 1039321 | Zbl 0684.54001
[8] Gaal L.S.: On the theory of $\mathbf (m,n)$-compact spaces. Pacific J. Math. 8 (1958), 721–734. DOI 10.2140/pjm.1958.8.721 | MR 0099644
[9] García-Ferreira S.: On FU($p$)-spaces and $p$-sequential spaces. Comment. Math. Univ. Carolin. 32 (1991), 161–171. MR 1118299 | Zbl 0789.54032
[10] García-Ferreira S.: Some generalizations of pseudocompactness. Papers on General Topology and Applications (Flushing, NY, 1992), Ann. New York Acad. Sci., 728, New York Acad. Sci., New York, 1994, pp. 22–31. MR 1467759 | Zbl 0911.54022
[11] Garcia-Ferreira S.: On two generalizations of pseudocompactness. Proceedings of the 14th Summer Conference on General Topology and its Applications (Brookville, NY, August 4–8, 1999), Topology Proc. 24 (2001), 149–172. MR 1876373 | Zbl 1026.54017
[12] Ginsburg J., Saks V.: Some applications of ultrafilters in topology. Pacific J. Math. 57 (1975), 403–418. DOI 10.2140/pjm.1975.57.403 | MR 0380736 | Zbl 0288.54020
[13] Glicksberg I.: Stone-Čech compactifications of products. Trans. Amer. Math. Soc 90 (1959), 369–382. MR 0105667 | Zbl 0089.38702
[14] Good C.: The Lindelöf property. in Encyclopedia of General Topology, edited by K.P. Hart, J. Nagata and J.E. Vaughan, Elsevier Science Publishers, Amsterdam, 2004, Chapter d-8, 182–184. MR 2049453
[15] Larson P.B.: Irreducibility of product spaces with finitely many points removed. Spring Topology and Dynamical Systems Conference, Topology Proc. 30 (2006), 327–333. MR 2280675 | Zbl 1128.54004
[16] Lipparini P.: Compact factors in finally compact products of topological spaces. Topology Appl. 153 (2006), 1365–1382. DOI 10.1016/j.topol.2005.04.002 | MR 2211205 | Zbl 1093.54001
[17] Lipparini P.: Some compactness properties related to pseudocompactness and ultrafilter convergence. Topology Proc. 40 (2012), 29–51. MR 2793281
[18] Lipparini P.: More generalizations of pseudocompactness. Topology Appl. 158 (2011), 1655–1666. DOI 10.1016/j.topol.2011.05.039 | MR 2812474 | Zbl 1239.54011
[19] Lipparini P.: Ordinal compactness. submitted, preprint available at arXiv:1012.4737v2 (2011).
[20] Lipparini P.: Products of sequentially pseudocompact spaces. arXiv:1201.4832.
[21] Scheepers M.: Combinatorics of open covers. I. Ramsey theory. Topology Appl. 69 (1996), 31–62. DOI 10.1016/0166-8641(95)00067-4 | MR 1378387 | Zbl 0848.54018
[22] Smirnov Y.M.: On topological spaces compact in a given interval of powers (Russian). Izvestiya Akad. Nauk SSSR, Ser. Mat. 14 (1950), 155–178. MR 0035004
[23] Stephenson R.M., Jr.: Initially $\kappa$-compact and related spaces. in Handbook of Set-theoretic Topology, edited by K. Kunen and J.E. Vaughan, North-Holland, Amsterdam, 1984, Chapter 13, pp. 603–632. MR 0776632 | Zbl 0588.54025
[24] Stephenson R.M., Jr.: Pseudocompact spaces. in Encyclopedia of General Topology, edited by K.P. Hart, J. Nagata and J.E. Vaughan, Elsevier Science Publishers, B.V., Amsterdam, 2004, Chapter d-07, pp. 177–181. Zbl 0804.54004
[25] Vaughan J.E.: Some recent results in the theory of $[a,b]$-compactness. in TOPO 72–General Topology and its Applications (Proc. Second Pittsburgh Internat. Conf., Pittsburgh, Pa., 1972; dedicated to the memory of Johannes H. de Groot), Lecture Notes in Math., 378, Springer, Berlin, 1974, pp. 534–550. DOI 10.1007/BFb0068506 | MR 0367928 | Zbl 0297.54021
[26] Vaughan J.E.: Some properties related to $[a,b]$-compactness. Fund. Math. 87 (1975), 251–260. MR 0380732
[27] Vaughan J.E.: Countably compact and sequentially compact spaces. in Handbook of Set-theoretic Topology, edited by K. Kunen and J.E. Vaughan, North-Holland, Amsterdam, 1984, Chapter 12, pp. 569–602. MR 0776631 | Zbl 0562.54031
[28] Vaughan J.E.: Countable compactness. in Encyclopedia of General Topology, edited by K.P. Hart, J. Nagata and J.E. Vaughan, Elsevier Science Publishers, Amsterdam, 2004, Chapter d-6, 174–176. MR 2049453 | Zbl 0984.54027

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