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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
##### Keywords:
relation of a given type; Čech topology; ternary relations
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##### References:
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