Products of derived structures on topological spaces

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• 作者：M.R. Burke^a ; ¹ ; ^{burke@upei.ca" class="auth_mail" title="E-mail the corresponding author} ; N.D. Macheras^b ; ^{macheras@unipi.gr" class="auth_mail" title="E-mail the corresponding author} ; W. Strauss^c ; ^{wekari@gmx.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：54E52 ; 54B10 ; 06E05 ; 60B05 ; 28A51 ; 28A12
• 刊名：Topology and its Applications
• 出版年：2016
• 出版时间：15 March 2016
• 年：2016
• 卷：201
• 期：Complete
• 页码：247-268
• 全文大小：593 K

文摘

We consider topological spaces X   equipped with an algebra 05714&_mathId=si1.gif&_user=111111111&_pii=S0166864115005714&_rdoc=1&_issn=01668641&md5=11a2ebf38e3a6a3bc9a0ae9944b56dba" title="Click to view the MathML source">A$\mathcal{A}$ of subsets of X   and an ideal 05714&_mathId=si170.gif&_user=111111111&_pii=S0166864115005714&_rdoc=1&_issn=01668641&md5=a5584db63deeb6364a677e0d15460ea6" title="Click to view the MathML source">I$\mathcal{I}$ of 05714&_mathId=si1.gif&_user=111111111&_pii=S0166864115005714&_rdoc=1&_issn=01668641&md5=11a2ebf38e3a6a3bc9a0ae9944b56dba" title="Click to view the MathML source">A$\mathcal{A}$. Motivated by the example of the Jordan measurable subsets of 05714&_mathId=si12.gif&_user=111111111&_pii=S0166864115005714&_rdoc=1&_issn=01668641&md5=1222f056d399f3607cc0cc0684356e0b" title="Click to view the MathML source">R$\mathbb{R}$, we consider the derived structure   obtained by replacing 05714&_mathId=si1.gif&_user=111111111&_pii=S0166864115005714&_rdoc=1&_issn=01668641&md5=11a2ebf38e3a6a3bc9a0ae9944b56dba" title="Click to view the MathML source">A$\mathcal{A}$ by the algebra 05714&_mathId=si5.gif&_user=111111111&_pii=S0166864115005714&_rdoc=1&_issn=01668641&md5=6f98f004cec0ffd730326bf774a7e9c7" title="Click to view the MathML source">∂A={E∈A:∂E∈I}$\partial \mathcal{A}=\{E\in \mathcal{A}:\partial E\in \mathcal{I}\}$ of sets with negligible boundaries, and 05714&_mathId=si170.gif&_user=111111111&_pii=S0166864115005714&_rdoc=1&_issn=01668641&md5=a5584db63deeb6364a677e0d15460ea6" title="Click to view the MathML source">I$\mathcal{I}$ by 05714&_mathId=si6.gif&_user=111111111&_pii=S0166864115005714&_rdoc=1&_issn=01668641&md5=888b076306390ad5b33880f480e9b285" title="Click to view the MathML source">∂I=I∩∂A$\partial I=\mathcal{I}\cap \partial \mathcal{A}$. In a previous paper by M.R. Burke et al. (2012) [7], the authors classified these derived structures (under some assumptions) and computed densities for them. In the present paper, we extend that work in the context of products of derived structures. We study in greater detail the box cross product 05714&_mathId=si545.gif&_user=111111111&_pii=S0166864115005714&_rdoc=1&_issn=01668641&md5=4387f4de0994a2bc038ead6949730781" title="Click to view the MathML source">γ⊠δ$\gamma ⊠\delta$ of two set maps 05714&_mathId=si541.gif&_user=111111111&_pii=S0166864115005714&_rdoc=1&_issn=01668641&md5=d998a3f4e47f8ac516ff15924ed419ec" title="Click to view the MathML source">γ∈P(X)^A$\gamma \in \mathcal{P}{(X)}^{\mathcal{A}}$, 05714&_mathId=si553.gif&_user=111111111&_pii=S0166864115005714&_rdoc=1&_issn=01668641&md5=d6fc96677008803e507525bf39da386f" title="Click to view the MathML source">δ∈P(Y)^B$\delta \in \mathcal{P}{(Y)}^{\mathcal{B}}$ introduced in joint work of the authors with K. Musiał (2009) [6], examining when it preserves densities and other types of liftings. For preservation of monotonicity, we introduce a variation on the localization property of ideals which is well-known for the meager ideal. An examination of skew products provides a class of structures to which our results apply.