Equality of dimensions for some paracompact \(\sigma \)-spaces. (English. Russian original) Zbl 1520.54017

Leibo, I. M.

Math. Notes 113, No. 4, 488-501 (2023); translation from Mat. Zametki 113, No. 4, 499-516 (2023).

A.V. Arhangel’skiǐ posed the following question: Do the dimensions of a first-countable regular space with a countable network coincide? In this paper, the author gives a partial positive answer to Arhangel’skiǐ’s question; namely, it is proved that \(\mbox{Ind}\,X=\dim X=\mbox{ind}\,X\) for any first-countable paracompact \(\sigma\)-space \(X\) with a separately continuous semi-metric. Therefore, the equality of the dimensions for first-countable stratifiable spaces with a separately continuous semi-metric is obtained. The following question posed by Arhangel’skiǐ is open: Is it true that \(\mbox{Ind}\,X=\dim X\) for any stratifiable topological group \(X\)?

Reviewer: Shou Lin (Ningde)

MSC:

54F45	Dimension theory in general topology
54D20	Noncompact covering properties (paracompact, Lindelöf, etc.)

Keywords:

dimension; network; \( \sigma \)-space; stratifiable space

References:

[1]	Aleksandrov, P. S.; Pasynkov, B. A., Introduction to Dimension Theory (1973), Moscow: Nauka, Moscow · Zbl 0272.54028
[2]	Arkhangel’skii, A. V., Mappings and spaces, Russian Math. Surveys, 21, 4, 115-162 (1966) · Zbl 0171.43603 · doi:10.1070/RM1966v021n04ABEH004169
[3]	Arkhangel’skii, A. V., Classes of topological groups, Russian Math. Surveys, 36, 3, 151-174 (1981) · Zbl 0488.22001 · doi:10.1070/RM1981v036n03ABEH004249
[4]	Leibo, I. M., On closed images of metric spaces, Soviet Math. Dokl., 16, 1292-1295 (1975) · Zbl 0329.54034
[5]	Leibo, I. M., On the dimensions of certain spaces, Soviet Math. Dokl., 25, 20-22 (1982) · Zbl 0504.54040
[6]	Borges, C. J. R., On continuously semimetrizable and stratifiable spaces, Proc. Amer. Math. Soc., 24, 2, 193-196 (1970) · Zbl 0185.26501 · doi:10.1090/S0002-9939-1970-0250266-7
[7]	Leibo, I. M., On the dimension of some semi-metric spaces, Conference on Set-Theoretic Topology and Topological Algebra, 36-37 (2018)
[8]	Leibo, I. M., On the dimension of preimages of certain paracompact spaces, Math. Notes, 103, 3, 405-414 (2018) · Zbl 1398.54055 · doi:10.1134/S0001434618030070
[9]	Engelking, R., Dimension Theory (1978), Amsterdam: North-Holland, Amsterdam · Zbl 0401.54029
[10]	Lin, Sh.; Yun, Z., Generalized Metric Spaces and Mappings (2016), Paris: Atlantis Press, Paris · Zbl 1366.54001 · doi:10.2991/978-94-6239-216-8
[11]	Oka, S., Dimension of stratifiable spaces, Trans. Amer. Math. Soc., 275, 1, 231-243 (1983) · Zbl 0508.54026 · doi:10.1090/S0002-9947-1983-0678346-6
[12]	Dugundji, J., An extension of Tietze’s theorem, Pacific J. Math., 1, 3, 353-367 (1951) · Zbl 0043.38105 · doi:10.2140/pjm.1951.1.353
[13]	Reed, G. M., Concerning first countable spaces, Fund. Math., 74, 1, 161-169 (1972) · Zbl 0227.54019 · doi:10.4064/fm-74-3-161-169
[14]	Ito, M., \(M_3\)-spaces whose every point has a closure preserving outer base are \(M_1\), Topology Appl., 19, 1, 65-69 (1985) · Zbl 0567.54012 · doi:10.1016/0166-8641(85)90085-9