Limit structures and many valued topologies
Let \({\mathcal F}(X)\) denote the set of all the filters on a set \(X\) and for each \(x\in X\), let \(\widehat{x}\) denote the principal filter generated by \(\{x\}\). A limit structure on \(X\) is a subset \(\mathcal T\) of \({\mathcal F}(X)\times X\) such that (1) \((\widehat{x}, x)\in \mathcal T\) for all \(x\in X\); (2) \((F, x)\in\mathcal T\) and \(F\subseteq G\) implies \((G, x)\in\mathcal T\) and (3) \((F, x)\in\mathcal T\) and \((G, x)\in\mathcal T\) implies \((F\cap G, x)\in\mathcal T\). \textbf{Lim} denotes the category of limit spaces. Let \((H, \leq)\) be a complete Heyting algebra, i.e., \((H, \leq)\) is a complete lattice and the binary meet operation \(\wedge\) is distributive over arbitrary joins \(\bigvee\). \(H\)-\textbf{TOP} denotes the category of \(H\)-valued topological spaces. In this paper, it is demonstrated that there exists an adjunction between the categories \textbf{Lim} and \(H\)-\textbf{TOP}. Roughly speaking, for every limit structure \(\mathcal T\) on a set \(X\), there is a corresponding \(H\)-valued topology on \(X\); and conversely, for each \(H\)-valued topology \(\tau\) on \(X\), there is a limit structure on \(X\) closely related to \(\tau\). Making use of this adjunction, it is showed that there is a Boolean-valued topology on the space of Borel measurable, almost everywhere (with respect to the Lebesgue measure) defined functions from [0, 1] to the real line \(\mathbb{R}\) such that a sequence of functions \(\{f_n\}\) converges with respect to the limit structure corresponding to this Boolean-valued topology iff it converges pointwise almost everywhere.
scientific article; zbMATH DE number 1556931