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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 ($${\mathcal F},L)$$.

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