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Minimal structures, generalized topologies, and ascending systems should not be studied without generalized uniformities. (English) Zbl 1199.54164

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.

MSC:

54E15 Uniform structures and generalizations
54A05 Topological spaces and generalizations (closure spaces, etc.)
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