Seminormal functors and paranormality. (English. Russian original) Zbl 1518.54006
Mosc. Univ. Math. Bull. 78, No. 2, 100-104 (2023); translation from Vestn. Mosk. Univ., Ser. I 78, No. 2, 67-71 (2023).
Summary: It is known that if a space \(\mathcal{F}(X)\) is hereditarily paranormal for a paracompact \(p\)-space \(X\) and normal functor \(\mathcal{F}\) of degree \(\geqslant 3\) in the category \(\mathcal{P}\) of paracompact \(p\)-spaces and their perfect maps, then \(X\) is metrizable [A. P. Kombarov, Mosc. Univ. Math. Bull. 72, No. 5, 203–205 (2017; Zbl 1383.54024); translation from Vestn. Mosk. Univ., Ser. I 72, No. 5, 48–51 (2017)]. In this paper, a generalization of this theorem is proved for seminormal functors in the category \(\mathcal{P} \).
MSC:
54B30 | Categorical methods in general topology |
54D20 | Noncompact covering properties (paracompact, Lindelöf, etc.) |
54E35 | Metric spaces, metrizability |
Citations:
Zbl 1383.54024References:
[1] | Engelking, R., General Topology (1977), Warsaw: PWN-Polish Scientific Publishers, Warsaw · Zbl 0373.54002 |
[2] | Shchepin, E. V., Functors and uncountable powers of compacta, Russ. Math. Surv., 36, 1-71 (1981) · Zbl 0487.54011 · doi:10.1070/RM1981v036n03ABEH004247 |
[3] | Fedorchuk, V. V., On the Katetov cube theorem, Moscow Univ. Math. Bull., 44, 102-106 (1989) · Zbl 0698.54006 |
[4] | Dobrynina, M. A., On Fedorchuk’s normal functor theorem, Math. Notes, 90, 611 (2011) · Zbl 1284.54042 · doi:10.1134/S000143461109029X |
[5] | Kombarov, A. P., The weak form of normality, Moscow Univ. Math. Bull., 72, 203-205 (2017) · Zbl 1383.54024 · doi:10.3103/S0027132217050047 |
[6] | Nyikos, P., Problem section: Problem B, Topol. Proc., 9, 367 (1984) · Zbl 0576.54001 |
[7] | Ivanov, A. A., Normal functors and paranormality, Moscow Univ. Math. Bull., 76, 271-273 (2021) · Zbl 1487.54037 · doi:10.3103/S0027132221060048 |
[8] | A. V. Ivanov, ‘‘Katetov’s cube theorem and seminormal functors,’’ Uch. Zap. Petrozavod. Gos. Univ., No. 2, 104-108 (2012). |
[9] | A. V. Arkhangel’skii, ‘‘A class of spaces which contains all metric and all locally compact spaces,’’ Mat. Sb. 67(109) (1), 55-88 (1965). · Zbl 0127.13101 |
[10] | Kombarov, A. P., Paranormality in topological products, Math. Notes, 102, 431-433 (2017) · Zbl 1383.54015 · doi:10.1134/S0001434617090140 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.