Topological spaces with the Freese-nation property. II. (English) Zbl 1479.54052
This article is a continuation of the research of J. Bak and A. Kucharski [Ann. Math. Sil. 33, 41–54 (2019; Zbl 1429.54018)] where the FN-property and the FNS-property were introduced. Here the authors study, moreover, the \(\pi\)-FNS property. A topological space has the \(\pi\)-FNS property if there exists a \(\pi\)-base with the FNS property. It is proved, in particular, that compact Hausdorff spaces with the FNS property for some base consisting of co-zero sets are openly generated spaces, and spaces with the \(\pi\)-FNS property are skeletally generated. Moreover, examples of families without the FN property and without the FNS property are given.
Reviewer: Hans Weber (Udine)
MSC:
| 54D70 | Base properties of topological spaces |
| 54G20 | Counterexamples in general topology |
| 91A44 | Games involving topology, set theory, or logic |
| 54F99 | Special properties of topological spaces |
Keywords:
FNS property; \(\pi\)-FNS property; inverse limit; openly generated space; skeletally generated spaceCitations:
Zbl 1429.54018References:
| [1] | Bąk, J.; Kucharski, A., Topological spaces with the Freese-Nation property, Ann. Math. Sil., 33, 1, 41-54 (2019) · Zbl 1429.54018 |
| [2] | Bąk, J.; Kucharski, A., An internal characterization of complete regularity, Math. Slovaca, 70, 3, 775-777 (2020) · Zbl 1505.54035 |
| [3] | Daniels, P.; Kunen, K.; Zhou, H., On the open-open game, Fundam. Math., 145, 3, 205-220 (1994) · Zbl 0811.54008 |
| [4] | Engelking, R., General Topology (1989), Heldermann Verlag: Heldermann Verlag Berlin · Zbl 0684.54001 |
| [5] | Freese, R.; Nation, J. B., Projective lattices, Pac. J. Math., 75, 93-106 (1975) · Zbl 0382.06005 |
| [6] | Fuchino, S.; Koppelberg, S.; Shelah, S., Partial orderings with the weak Freese-Nation property, Ann. Pure Appl. Log., 80, 35-54 (1996) · Zbl 0968.03048 |
| [7] | Fuchino, S.; Soukup, L., More set-theory around the weak Freese-Nation property, Fundam. Math., 154, 159-176 (1997) · Zbl 0882.03046 |
| [8] | Haydon, R., On a problem of Pelczynski: Milutin spaces, Dugundji spaces and \(A E(0 - d i m)\), Stud. Math., 52, 23-31 (1974) · Zbl 0294.46016 |
| [9] | Heindorf, L.; Shapiro, L. B., Nearly Projective Boolean Algebras, Lecture Notes in Math., vol. 1596 (1994) · Zbl 0812.06007 |
| [10] | Kucharski, A.; Plewik, Sz., Inverse systems and I-favorable spaces, Topol. Appl., 156, 1, 110-116 (2008) · Zbl 1160.54007 |
| [11] | Kucharski, A.; Plewik, Sz.; Valov, V., Game theoretic approach to skeletally Dugundji and Dugundji spaces, Topol. Appl., 201, 206-216 (2016) · Zbl 1345.54013 |
| [12] | Kucharski, A.; Plewik, Sz.; Valov, V., Skeletally Dugundji spaces, Cent. Eur. J. Math., 11, 1949-1959 (2013) · Zbl 1296.54029 |
| [13] | Mioduszewski, J.; Rudolf, L., H-closed and extremally disconnected Hausdorff spaces, Diss. Math., 66 (1969) · Zbl 0204.22404 |
| [14] | Milovich, D., On the strong Freese-Nation property, Order, 34, 1, 91-111 (2017) · Zbl 1373.06016 |
| [15] | Milovich, D., The \((\lambda, \kappa)\)-Freese-Nation property for Boolean algebras and compacta, Order, 29, 361-379 (2012) · Zbl 1251.03052 |
| [16] | Pełczyński, A., Linear extensions, linear averagings, and their applications to linear topological classification of spaces of continuous functions, Diss. Math., 58, 1-89 (1968) · Zbl 0165.14603 |
| [17] | Porter, J. R.; Woods, G. R., Extensions and Absolutes of Hausdorff Spaces (1988), Springer-Verlag: Springer-Verlag New York · Zbl 0652.54016 |
| [18] | Shapiro, L., On spaces co-absolute with dyadic compacta, Dokl. Akad. Nauk SSSR, 293, 5, 1077-1081 (1987), (in Russian) |
| [19] | Shchepin, E. V., Topology of limit spaces with uncountable inverse spectra, Usp. Mat. Nauk, 31, 5(191), 191-226 (1976) · Zbl 0356.54026 |
| [20] | Shchepin, E. V., On κ-metrizable spaces, Izv. Akad. Nauk SSSR, Ser. Mat., 43, 2, 442-478 (1979), (Russian) · Zbl 0409.54040 |
| [21] | Shchepin, E. V., Functors and uncountable powers of compacta, Usp. Mat. Nauk, 36, 3(219), 3-62 (1981) · Zbl 0463.54009 |
| [22] | Tkachenko, M., Some results on inverse spectra II, Comment. Math. Univ. Carol., 22, 819-841 (1981) · Zbl 0494.54007 |
| [23] | Woods, R. G., A survey of absolutes of topological spaces, (Topological Structures II. Topological Structures II, Amsterdam. Topological Structures II. Topological Structures II, Amsterdam, Mathematische Centrum Tract, vol. 116 (1979)), 323-362 · Zbl 0481.54021 |
| [24] | Valov, V., I-favorable spaces: revisited, Topol. Proc., 51, 277-292 (2018) · Zbl 1386.54006 |
| [25] | Valov, V., External characterization of I-favorable spaces, Math. Balk., 25, 1-2, 61-78 (2011) · Zbl 1235.54009 |
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