Two weak forms of countability axioms in free topological groups

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• 作者：Fucai Lin^a ; ^{linfucai2008@aliyun.com" class="auth_mail" title="E-mail the corresponding author} ; Chuan Liu^b ; ^{liuc1@ohio.edu" class="auth_mail" title="E-mail the corresponding author} ; Jiling Cao^c ; ^{jiling.cao@aut.ac.nz" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 54H11 ; 22A05 ; secondary ; 54E20 ; 54E35 ; 54D50 ; 54D55
• 刊名：Topology and its Applications
• 出版年：2016
• 出版时间：1 July 2016
• 年：2016
• 卷：207
• 期：Complete
• 页码：96-108
• 全文大小：422 K

文摘

Given a Tychonoff space X  , let pan id="mmlsi1" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116300700&_mathId=si1.gif&_user=111111111&_pii=S0166864116300700&_rdoc=1&_issn=01668641&md5=9c75474d300a23914d2847c7260e8970" title="Click to view the MathML source">F(X)pan>pan class="mathContainer hidden">pan class="mathCode">$F(X)$pan>pan>pan> and pan id="mmlsi2" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116300700&_mathId=si2.gif&_user=111111111&_pii=S0166864116300700&_rdoc=1&_issn=01668641&md5=55a4bd18e1346bf3ed86cab61813dd59" title="Click to view the MathML source">A(X)pan>pan class="mathContainer hidden">pan class="mathCode">$A(X)$pan>pan>pan> be respectively the free topological group and the free Abelian topological group over X   in the sense of Markov. For every pan id="mmlsi16" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116300700&_mathId=si16.gif&_user=111111111&_pii=S0166864116300700&_rdoc=1&_issn=01668641&md5=2a8ec6d9e9ccb78de0959c8131bdf846" title="Click to view the MathML source">n∈Npan>pan class="mathContainer hidden">pan class="mathCode">$n\in \mathbb{N}$pan>pan>pan>, let pan id="mmlsi168" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116300700&_mathId=si168.gif&_user=111111111&_pii=S0166864116300700&_rdoc=1&_issn=01668641&md5=a824f0cce9c1271833bcc50e4c9f18ee" title="Click to view the MathML source">F_n(X)pan>pan class="mathContainer hidden">pan class="mathCode">$F_n(X)$pan>pan>pan> (resp. pan id="mmlsi362" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116300700&_mathId=si362.gif&_user=111111111&_pii=S0166864116300700&_rdoc=1&_issn=01668641&md5=ecd3b403c73bea656d859c2c8a790754" title="Click to view the MathML source">A_n(X)pan>pan class="mathContainer hidden">pan class="mathCode">$A_n(X)$pan>pan>pan>) denote the subspace of pan id="mmlsi1" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116300700&_mathId=si1.gif&_user=111111111&_pii=S0166864116300700&_rdoc=1&_issn=01668641&md5=9c75474d300a23914d2847c7260e8970" title="Click to view the MathML source">F(X)pan>pan class="mathContainer hidden">pan class="mathCode">$F(X)$pan>pan>pan> (resp. pan id="mmlsi2" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116300700&_mathId=si2.gif&_user=111111111&_pii=S0166864116300700&_rdoc=1&_issn=01668641&md5=55a4bd18e1346bf3ed86cab61813dd59" title="Click to view the MathML source">A(X)pan>pan class="mathContainer hidden">pan class="mathCode">$A(X)$pan>pan>pan>) that consists of words of reduced length at most n with respect to the free basis X  . In this paper, we discuss two weak forms of countability axioms in pan id="mmlsi1" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116300700&_mathId=si1.gif&_user=111111111&_pii=S0166864116300700&_rdoc=1&_issn=01668641&md5=9c75474d300a23914d2847c7260e8970" title="Click to view the MathML source">F(X)pan>pan class="mathContainer hidden">pan class="mathCode">$F(X)$pan>pan>pan> or pan id="mmlsi2" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116300700&_mathId=si2.gif&_user=111111111&_pii=S0166864116300700&_rdoc=1&_issn=01668641&md5=55a4bd18e1346bf3ed86cab61813dd59" title="Click to view the MathML source">A(X)pan>pan class="mathContainer hidden">pan class="mathCode">$A(X)$pan>pan>pan>, namely the csf-countability and snf-countability. We provide some characterizations of the csf-countability and snf  -countability of pan id="mmlsi1" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116300700&_mathId=si1.gif&_user=111111111&_pii=S0166864116300700&_rdoc=1&_issn=01668641&md5=9c75474d300a23914d2847c7260e8970" title="Click to view the MathML source">F(X)pan>pan class="mathContainer hidden">pan class="mathCode">$F(X)$pan>pan>pan> and pan id="mmlsi2" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116300700&_mathId=si2.gif&_user=111111111&_pii=S0166864116300700&_rdoc=1&_issn=01668641&md5=55a4bd18e1346bf3ed86cab61813dd59" title="Click to view the MathML source">A(X)pan>pan class="mathContainer hidden">pan class="mathCode">$A(X)$pan>pan>pan> for various classes of spaces X. In addition, we also study the csf-countability and snf  -countability of pan id="mmlsi168" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116300700&_mathId=si168.gif&_user=111111111&_pii=S0166864116300700&_rdoc=1&_issn=01668641&md5=a824f0cce9c1271833bcc50e4c9f18ee" title="Click to view the MathML source">F_n(X)pan>pan class="mathContainer hidden">pan class="mathCode">$F_n(X)$pan>pan>pan> or pan id="mmlsi362" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116300700&_mathId=si362.gif&_user=111111111&_pii=S0166864116300700&_rdoc=1&_issn=01668641&md5=ecd3b403c73bea656d859c2c8a790754" title="Click to view the MathML source">A_n(X)pan>pan class="mathContainer hidden">pan class="mathCode">$A_n(X)$pan>pan>pan>, for pan id="mmlsi6" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116300700&_mathId=si6.gif&_user=111111111&_pii=S0166864116300700&_rdoc=1&_issn=01668641&md5=ad59f805747acc895dbe4d583fbb358a" title="Click to view the MathML source">n=2,3,4pan>pan class="mathContainer hidden">pan class="mathCode">$n=2,3,4$pan>pan>pan>. Some results of Arhangel'skiı̌ in pan id="bbr0010">[1]pan> and Yamada in pan id="bbr0220">[20]pan> are generalized. An affirmative answer to an open question posed by Li et al. in pan id="bbr0110">[11]pan> is provided.