Three-space properties in paratopological groups

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• 作者：Manuel Ferná ; ndez^a ; ^{manuel.fernandezvillanueva@uacm.edu.mx} ; ^{mafevil5@gmail.com} ; Ivá ; n Sá ; nchez^b ; ^{isr.uami@gmail.com}
• 关键词：primary ; 54H11 ; 22A05 ; secondary ; 54B15 ; 54D30 ; 54E35
• 刊名：Topology and its Applications
• 出版年：2017
• 出版时间：1 February 2017
• 年：2017
• 卷：216
• 期：Complete
• 页码：74-78
• 全文大小：252 K
• 卷排序：216

文摘

We show that neither first-countable nor second-countable are three-space properties in the class of paratopological groups: we present a countable regular paratopological Abelian group H which contains a closed discrete subgroup F   such that H/FH/F is topologically isomorphic to the rational numbers with the Sorgenfrey topology and H is not first-countable. Also, we prove that if H is an invariant topological subgroup of a paratopological group G such that H   is second-countable and G/HG/H has countable network, then G has countable network as well (this answers a question posed in [12]). Hence if H is an invariant topological subgroup of a first-countable paratopological group G such that H   and G/HG/H are second-countable, then so is G.