Ideal QN-spaces

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• 作者：Jaroslav &Scaron ; upina¹ ; ^{jaroslav.supina@upjs.sk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Ideal convergence ; IIQN-space ; Weak P-ideal
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：1 March 2016
• 年：2016
• 卷：435
• 期：1
• 页码：477-491
• 全文大小：439 K

文摘

We continue to investigate an ideal version of QN-space, a X15009786&_mathId=si1.gif&_user=111111111&_pii=S0022247X15009786&_rdoc=1&_issn=0022247X&md5=a9c106026cfa0403cea4e549c6851df3" title="Click to view the MathML source">J$\mathcal{J}$QN-space, introduced by P. Das and D. Chandra [8]. Following R. Filipów and M. Staniszewski [15], we show that an ideal J$\mathcal{J}$ on ω   contains an isomorphic copy of the ideal Fin×Fin$\mathrm{Fin}\times \mathrm{Fin}$ on ω×ω$\omega \times \omega$ if and only if every topological space is a J$\mathcal{J}$QN-space. If J$\mathcal{J}$ does not contain an isomorphic copy of the ideal Fin×Fin$\mathrm{Fin}\times \mathrm{Fin}$ then the Baire space ${}^{\omega }\omega$ is not a J$\mathcal{J}$QN-space. However, if p=c$\mathfrak{p}=\mathfrak{c}$ then there is an ideal J$\mathcal{J}$ not containing an isomorphic copy of the ideal Fin×Fin$\mathrm{Fin}\times \mathrm{Fin}$ and there is a J$\mathcal{J}$QN-space which is not a QN-space. We prove few results related to an ideal version of an S₁(Γ,Γ)${\mathrm{S}}_1(\mathrm{\Gamma },\mathrm{\Gamma })$-space. Indeed, we show that there is no ideal J$\mathcal{J}$ such that the notion of an S₁(Γ,J-Γ)${\mathrm{S}}_1(\mathrm{\Gamma },\mathcal{J}\text{-}\mathrm{\Gamma })$-space is trivial. Consequently the ideal version of Scheepers' Conjecture does not hold for ideals containing an isomorphic copy of Fin×Fin$\mathrm{Fin}\times \mathrm{Fin}$.