On the Dini and Stone-Weierstrass properties in pointfree topology

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• 作者：David Holgate^a ; ^{dholgate@uwc.ac.za" class="auth_mail" title="E-mail the corresponding author} ; Jacques Masuret^b ; ^{jmasuret@sun.ac.za" class="auth_mail" title="E-mail the corresponding author} ; Mark Sioen^c ; ^{msioen@vub.ac.be" class="auth_mail" title="E-mail the corresponding author}
• 关键词：06D22 ; 06F25 ; 54C30 ; 54A05 ; 54D99
• 刊名：Topology and its Applications
• 出版年：2016
• 出版时间：1 March 2016
• 年：2016
• 卷：200
• 期：Complete
• 页码：160-175
• 全文大小：386 K

文摘

For a topological space X, it is customary to equip the f  -ring C(X)$C(X)$ or its bounded part C^⁎(X)$C^{⁎}(X)$ with the well-known topologies or uniformities of uniform convergence or pointwise convergence, which thank their importance to several fundamental theorems like a.o. the Stone–Weierstrass theorem or Dini's theorem. These theorems are classically proved subject to possible supplementary (often compactness) conditions on X. Alternatively, one can also in many cases characterize exactly those X for which the conclusion of such a theorem holds, i.e. those X that have the Stone–Weierstrass or Dini property. In pointfree topology, for a frame L  , one encounters as a counterpart to C(X)$C(X)$ (resp. C^⁎(X)$C^{⁎}(X)$), the well-studied f  -ring RL$\mathcal{R}L$ of real-valued continuous functions on L   and its bounded part R^⁎L${\mathcal{R}}^{⁎}L$. A pointfree Stone–Weierstrass theorem has been proved in B. Banaschewski  and . It is the aim of this note to discuss some topological properties of the f  -ring RL$\mathcal{R}L$ or its bounded part which are pointfree counterparts of the Stone–Weierstrass and Dini-type properties for spaces.