Száz, Árpád Minimal structures, generalized topologies, and ascending systems should not be studied without generalized uniformities. (English) Zbl 1199.54164 Filomat 21, No. 1, 87-97 (2007). A minimal structure on a set \(X\) is a subfamily \(\mathcal A\) of the power set \(\mathcal P(X)\) containing the empty set and \(X\). A minimal structure closed under arbitrary unions is a generalized topology or supratopology. An ascending family \(\mathcal A\subset\mathcal P(X)\) (for which \(A\in\mathcal A\) and \(A\subset B\subset X\) imply \(B\in\mathcal A\)) is a stack. The author proves that every minimal structure (resp., generalized topology, ascending system (proper stack)) on a set \(X\) can be obtained by means of some nonempty relator \(\mathcal R\) (a nonempty family of binary relations on \(X\)) introduced by the author in [Acta Math. Hung. 50, 177–201 (1987; Zbl 0643.54033)]. Pervin relations [W. J. Pervin, Math. Ann. 147, 316–317 (1962; Zbl 0101.40501)] are used to construct this relator. The relator space \(X(\mathcal R)=(X,\mathcal R)\) is called a generalized uniformity. Reviewer: Mila Mršević (Beograd) Cited in 12 Documents MSC: 54E15 Uniform structures and generalizations 54A05 Topological spaces and generalizations (closure spaces, etc.) Keywords:relator spaces; proper stacks Citations:Zbl 0643.54033; Zbl 0101.40501 PDFBibTeX XMLCite \textit{Á. Száz}, Filomat 21, No. 1, 87--97 (2007; Zbl 1199.54164) Full Text: DOI