S₂ and the Fréchet property of free topological groups

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• 作者：Zhangyong Cai^a ; ^{zycaigxu2002@126.com" class="auth_mail" title="E-mail the corresponding author} ; Shou Lin^b ; ^{shoulin60@163.com" class="auth_mail" title="E-mail the corresponding author} ; Chuan Liu^c ; ^{liuc1@ohio.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：22A05 ; 54A20 ; 54D50 ; 54D55
• 刊名：Topology and its Applications
• 出版年：2016
• 出版时间：15 May 2016
• 年：2016
• 卷：204
• 期：Complete
• 页码：103-111
• 全文大小：348 K

文摘

Let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116001164&_mathId=si1.gif&_user=111111111&_pii=S0166864116001164&_rdoc=1&_issn=01668641&md5=e8b782eb1a49bb14900a00112ae74147" title="Click to view the MathML source">F(X)class="mathContainer hidden">class="mathCode">$F(X)$ denote the free topological group over a Tychonoff space X  , class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116001164&_mathId=si2.gif&_user=111111111&_pii=S0166864116001164&_rdoc=1&_issn=01668641&md5=47cd0b06199ddeef1304a4629683d16f" title="Click to view the MathML source">F_n(X)class="mathContainer hidden">class="mathCode">$F_n(X)$ denote the subspace of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116001164&_mathId=si1.gif&_user=111111111&_pii=S0166864116001164&_rdoc=1&_issn=01668641&md5=e8b782eb1a49bb14900a00112ae74147" title="Click to view the MathML source">F(X)class="mathContainer hidden">class="mathCode">$F(X)$ that consists of all words of reduced length ≤n with respect to the free basis X for every non-negative integer n   and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116001164&_mathId=si3.gif&_user=111111111&_pii=S0166864116001164&_rdoc=1&_issn=01668641&md5=f9fd571c14c3e622a86e036cd10d6596" title="Click to view the MathML source">E_n(X)=F_n(X)∖F_n−1(X)class="mathContainer hidden">class="mathCode">$E_n(X)=F_n(X)\setminus F_{n-1}(X)$ for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116001164&_mathId=si4.gif&_user=111111111&_pii=S0166864116001164&_rdoc=1&_issn=01668641&md5=e2e30a8bebf725382c6d0f9f4c4c13ea" title="Click to view the MathML source">n≥1class="mathContainer hidden">class="mathCode">$n\ge 1$. In this paper, we study topological properties of free topological groups in terms of Arens' space class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116001164&_mathId=si37.gif&_user=111111111&_pii=S0166864116001164&_rdoc=1&_issn=01668641&md5=6d53b9bf560101b946dd389cb84cd1f9" title="Click to view the MathML source">S₂class="mathContainer hidden">class="mathCode">$S_2$. The following results are obtained.

(1) If the free topological group class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116001164&_mathId=si1.gif&_user=111111111&_pii=S0166864116001164&_rdoc=1&_issn=01668641&md5=e8b782eb1a49bb14900a00112ae74147" title="Click to view the MathML source">F(X)class="mathContainer hidden">class="mathCode">$F(X)$ over a Tychonoff space X   contains a non-trivial convergent sequence, then class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116001164&_mathId=si1.gif&_user=111111111&_pii=S0166864116001164&_rdoc=1&_issn=01668641&md5=e8b782eb1a49bb14900a00112ae74147" title="Click to view the MathML source">F(X)class="mathContainer hidden">class="mathCode">$F(X)$ contains a closed copy of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116001164&_mathId=si37.gif&_user=111111111&_pii=S0166864116001164&_rdoc=1&_issn=01668641&md5=6d53b9bf560101b946dd389cb84cd1f9" title="Click to view the MathML source">S₂class="mathContainer hidden">class="mathCode">$S_2$, equivalently, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116001164&_mathId=si1.gif&_user=111111111&_pii=S0166864116001164&_rdoc=1&_issn=01668641&md5=e8b782eb1a49bb14900a00112ae74147" title="Click to view the MathML source">F(X)class="mathContainer hidden">class="mathCode">$F(X)$ contains a closed copy of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116001164&_mathId=si12.gif&_user=111111111&_pii=S0166864116001164&_rdoc=1&_issn=01668641&md5=6d84ff01881868d45045d6ec53398fe2" title="Click to view the MathML source">S_ωclass="mathContainer hidden">class="mathCode">$S_{\omega }$, which extends [6, Theorem 1.6].

(2) Let X   be a topological space and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116001164&_mathId=si141.gif&_user=111111111&_pii=S0166864116001164&_rdoc=1&_issn=01668641&md5=fbf53838a760b311ec787d59ca900af0" title="Click to view the MathML source">A={n₁,...,n_i,...}class="mathContainer hidden">class="mathCode">$A=\{n_1,...,n_i,...\}$ be an infinite subset of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116001164&_mathId=si142.gif&_user=111111111&_pii=S0166864116001164&_rdoc=1&_issn=01668641&md5=e3b5c2defd027a9ad9a426b148ee0f0f" title="Click to view the MathML source">Nclass="mathContainer hidden">class="mathCode">$\mathbb{N}$. If class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116001164&_mathId=si166.gif&_user=111111111&_pii=S0166864116001164&_rdoc=1&_issn=01668641&md5=8e203eb0c76eb16df43bbcf972206351" title="Click to view the MathML source">C=⋃_i∈NE_ni(X)class="mathContainer hidden">class="mathCode">$C={\bigcup }_{i\in \mathbb{N}}{E}_{n_i}(X)$ is κ  -Fréchet–Urysohn and contains no copy of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116001164&_mathId=si37.gif&_user=111111111&_pii=S0166864116001164&_rdoc=1&_issn=01668641&md5=6d53b9bf560101b946dd389cb84cd1f9" title="Click to view the MathML source">S₂class="mathContainer hidden">class="mathCode">$S_2$, then X is discrete, which improves [15, Proposition 3.5].

(3) If X is a μ  -space and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116001164&_mathId=si10.gif&_user=111111111&_pii=S0166864116001164&_rdoc=1&_issn=01668641&md5=8b2859708678b29663218ea35935db96" title="Click to view the MathML source">F₅(X)class="mathContainer hidden">class="mathCode">$F_5(X)$ is Fréchet–Urysohn, then X is compact or discrete, which improves [15, Theorem 2.4].

At last, a question posed by K. Yamada is partially answered in a shorter alternative way by means of a Tanaka's theorem concerning Arens' space class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116001164&_mathId=si37.gif&_user=111111111&_pii=S0166864116001164&_rdoc=1&_issn=01668641&md5=6d53b9bf560101b946dd389cb84cd1f9" title="Click to view the MathML source">S₂class="mathContainer hidden">class="mathCode">$S_2$.