Topological projective planes and uniform valuations

Abstract

The valuation topology of any uniformly valued ternary field (K, T, v) can be extended to the projective plane H over (K, T) making it a topological projective plane in the sense of Salzmann. Appealing to PrieB-Crampe's celebrated fixed point theorem for ultrametric spaces, our result allows us to present a wide variety of new, totally disconnected, compact and non-compact topological projective planes. (1991). 51H 10, 51 A10, 17A80, 12K99. Mathematics Subject Classifications* Asusuallet a--}-b := T(1, a, b), ab := T(a, b, 0), K* := K\{O},andwdtea-b, -b, a/c, and c\a for the elements defined by (a -b) -t-b = a, (-b) -t-b = 0, (a/c)c = a, and c(c\a) = a respectively (a, b, c E K, c ¢ 0). ** In contrast to [8], we require a uniform valuation to be onto, excluding the trivial valuation from our considerations.