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Extension of the ring topology of a \(\sigma\)-bounded field on its simple transcendental extension. (Russian) Zbl 0631.12020
Main result: Let \(\tau\) be a ring topology on a field R such that R is a union of a countable family of its bounded subsets (that is, R is \(\sigma\)-bounded; a set \(\Gamma\) is called bounded if for any neighbourhood U of zero there exists a neighbourhood V of zero such that \(\Gamma\cdot V\subset U\) and \(V\cdot \Gamma \subset U)\). Then there exists a topology \(\tau\) ’ on the simple transcendental extension R[x] of the field R making R[x] a \(\sigma\)-bounded topological ring and extending the topology \(\tau\) (that is, \(\tau '|_ R=\tau).\)
This result strengthens some earlier results due to K.-P. Podewski [Proc. Am. Math. Soc. 39, 33-38 (1973; Zbl 0268.12102)] and the author himself [Algebra Logika 20(50), 511-521 (1981; Zbl 0504.12023)].
The proof is constructive; basic neighbourhoods of zero in R[x] are described in a direct, but complicated and purely technical (“hard”) manner. It is interesting whether this proof can be “softened”.
Reviewer: V.G.Pestov

MSC:
12J99 Topological fields
13J99 Topological rings and modules
54H13 Topological fields, rings, etc. (topological aspects)
12F20 Transcendental field extensions
12E10 Special polynomials in general fields
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