zbMATH — the first resource for mathematics

Relations and topologies. (English) Zbl 0797.04002
By a relation of type \(\alpha\) (where \(\alpha> 0\) is an ordinal) on a set \(G\) the author means a set of sequences of type \(\alpha\) of elements of \(G\), by a topology on \(G\) a topology in Čech’s sense, i.e. a mapping \(u: \exp G\to \exp G\) with \(u\emptyset= \emptyset\), \(X\subseteq uX\), \(X\subseteq Y\Rightarrow uX\subseteq uY\). He assigns to any relation \(R\) on \(G\) a topology \(u_ R\) on \(G\) by \(u_ R X= X\cup\{x\in G\); there is \((x_ i; i<\alpha)\in R\) and an ordinal \(i_ 0\), \(0< i_ 0< \alpha\) with \(x= x_{i_ 0}\) and \(x_ i\in X\) for \(i< i_ 0\}\) and to any topology \(u\) on \(G\) a relation \(R_ u\) of type \(\alpha\) on \(G\) by \((x_ i; i<\alpha)\in R_ u\Leftrightarrow x_{i_ 0}\in u\{x_ i; i< i_ 0\}\) for any \(0< i_ 0< \alpha\).
Some properties of this assigning are derived, e.g., conditions on a topology \(v\) are given under which \(v= u_{R_ v}\). The above constructions are then applied to ternary relations with some special properties.
Reviewer: V.Novák (Brno)

03E20 Other classical set theory (including functions, relations, and set algebra)
54A05 Topological spaces and generalizations (closure spaces, etc.)
08A02 Relational systems, laws of composition
Full Text: EuDML
[1] Bar-Hillel, Y., Fraenkel, A. A., and Lévy, A.: Foundations of Set Theory. North Holand, Amsterdam, 1973.
[2] Bourbaki, N.: Topologie générale. Eléments de Mathématique, I. part., livre III, Paris, 1940. · Zbl 0026.43101
[3] Čech, E.: Topological spaces. Topological papers of Eduard Čech, Academia, Prague, 1968. · Zbl 0141.39401
[4] Čech, E.: Topological Spaces (Revised by Z. Frolík and M. Katětov). Academia, Prague, 1966.
[5] Mac Lane, S.: Categories for the Working Mathematician. Springer-Verlag, New York-Heidelberg-Berlin, 1971. · Zbl 0232.18001
[6] Novák, V., and Novotný, M.: Transitive ternary relations and quasiorderings. Arch. Math. Brno 25 (1989), 5-12. · Zbl 0714.06001 · eudml:18252
[7] Novotný, M.: Ternary structures and grupoids. Czech. Math. J. 41 (1991), 90-98. · Zbl 0790.20090 · eudml:13903
[8] Šlapal, J.: On closure operations induced by binary relations. Rev. Roum. Math. Pures et Appl. 33 (1988), 623-630. · Zbl 0658.54002
[9] Šlapal, J.: Relations of type \( \)Zeitschr. für Math. Logik und Grundl. der Math. 34 1988 563-573. · Zbl 0668.04002 · doi:10.1002/malq.19880340608
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.