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.

中文翻译：

---

ON -不利空间

为了研究一个超拓扑群何时变成一个拓扑群，Arhangel'skii等。['拓扑游戏和组上的拓扑'，数学。磨碎的。 8(2010), 1-19] 介绍了$(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$-不利的空间。我们表明，每$\unicode[STIX]{x1D707}$-完成（或正常）$(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$-不利的半拓扑群是一个拓扑群。我们证明a的乘积$(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$- 不利的空间和强烈的 Fréchet$(\unicode[STIX]{x1D6FC},G_{\unicode[STIX]{x1D6F1}})$- 有利的空间是$(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$-不利。我们还表明，连续封闭的不可约映射保留了$(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$- 双向不利。