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Statistical linear spaces. I: Properties of \(\epsilon\),\(\eta\)-topology. (English) Zbl 0542.46006
The definition of the statistical linear space in the Menger sense (SLM- space) (S,\({\mathcal J},T)\) is given in this paper. The \(\epsilon\),\(\eta\)- topology is introduced and fundamental properties of SLM-spaces as linear topological spaces are investigated.
Using the notion of F-convergent sequence it is possible to introduce a topology into S, named th \(\epsilon\),\(\eta\)-topology. Under continuity of the t-norm (S,\({\mathcal J},T)\) is a Hausdorff linear topological space with a countable base of neighbourhoods of the null element in S. If (S,\({\mathcal J},T)\) is finite-dimensional then the \(\epsilon\),\(\eta\)-topology is normable and equivalent to Euclidean topology. Some properties of the \(\epsilon\),\(\eta\)-topology can be investigated by a mapping \({\mathcal J}:S\to({\mathcal F},L),\) where L isLévy’s metric in \({\mathcal F}\). This mapping \({\mathcal J}\) is uniformly continuous and a subset \(K\subset(S,{\mathcal J},T)\) is \(\epsilon\),\(\eta\)-bounded if and only if \({\mathcal J}(K)\) is compact in (\({\mathcal F},L)\). As a space \((S,{\mathcal J},\min)\) with the strongest t-norm \(\min (\cdot,\cdot)\) is locally convex and hence the \(\epsilon\),\(\eta\)-topology is normable if and only if there exists an \(\epsilon\),\(\eta\)-neighbourhood \(O(\epsilon\),\(\eta)\) with compact image in (\({\mathcal F},L)\).

46A99 Topological linear spaces and related structures
54A40 Fuzzy topology
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