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ON $(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$-UNFAVOURABLE SPACES

Part of: Topological and differentiable algebraic systems Spaces with richer structures Fairly general properties

Published online by Cambridge University Press:  16 March 2020

HANFENG WANG Open the ORCID record for HANFENG WANG [Opens in a new window] ,
WEI HE Open the ORCID record for WEI HE [Opens in a new window]  and
JING ZHANG Open the ORCID record for JING ZHANG [Opens in a new window]
HANFENG WANG*
Affiliation:
Department of Mathematics, Shandong Agricultural University, Taian 271018, China email whfeng@sdau.edu.cn
WEI HE
Affiliation:
Institute of Mathematics, Nanjing Normal University, Nanjing 210046, China email weihe@njnu.edu.cn
JING ZHANG
Affiliation:
School of Mathematics and Statistics, Minnan Normal University, Zhangzhou 363000, China email zhangjing86@126.com
*
✉whfeng@sdau.edu.cn
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Abstract

To study when a paratopological group becomes a topological group, Arhangel’skii et al. [‘Topological games and topologies on groups’, Math. Maced. 8 (2010), 1–19] introduced the class of $(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$-unfavourable spaces. We show that every $\unicode[STIX]{x1D707}$-complete (or normal) $(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$-unfavourable semitopological group is a topological group. We prove that the product of a $(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$-unfavourable space and a strongly Fréchet $(\unicode[STIX]{x1D6FC},G_{\unicode[STIX]{x1D6F1}})$-favourable space is $(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$-unfavourable. We also show that continuous closed irreducible mappings preserve the $(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$-unfavourableness in both directions.

Keywords

G𝛱 game( 𝛽, G𝛱)-unfavourablestrategytopological group

MSC classification

Primary: 22A05: Structure of general topological groups
Secondary: 54D40: Remainders 54E35: Metric spaces, metrizability
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 102 , Issue 3 , December 2020 , pp. 439 - 450
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

Project supported by SDNSF (No. ZR2018MA013) and NSFC (Nos. 11571175, 11801254).

References

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