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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)

MSC:
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
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References:
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