On non-archimedean Structures of D. KLAUA

✍ Scribed by Andrea Cantini

• Publisher: John Wiley and Sons
• Year: 1979
• Tongue: English
• Weight: 390 KB
• Volume: 87
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19790870105

✦ Synopsis

In this paper we study local properties of systems C , of real transfinite numbers (KLAUA [l], [2], [3]), both from an algebraic and a topological point of view. In the first two sections, we consider C, as a valued field, endowed with a nonnrchimedean intrinsic valuation ; we characterize its residue class field, (which is isomorphic with the field of rational numbers) and its valuation group. Secondly we investigate the structure of its convex subgroups. In the last section we apply non-standard topological notions (ZAKON [4]) to the theory of hcompoiientv ' (KLAUA [2]) and generalize it. We obtain further a theorem of "global" pseudometrizability, which extends a preceding result of local metrizability in KLAUA [2].
Remark. The paper is not self-contained; it presupposes acquaintance with KLAUA [2] and some basic notions about valued fields (see for instance Born-

BAEI [5]

).
1.
-. We first introduce a bit of terminology and definitions.

Definitions

. 1 . l . 1. Suppose a, b E C,. Then 1.1.2. a m h b iff there is a natural number kwO, such that la-bl-ckh, where 1.1.3. aEP iff there exists a natural number k such that l/k-=IalO, hEC,. finite).

1. This paper contains some results of one chapter of author's "laurea" thesis, (Florence, October 1974).