Locally unbounded topological fields with topological nilpotents

Abstract

Abstract. We construct some locally unbounded topological fields having topologically nilpotent elements; this answers a question of Heine. The underlying fields are subfields of fields of formal power series. In particular, we get a locally unbounded topological field for which the set of topologically nilpotent elements is an open additive subgroup. We also exhibit a complete locally unbounded topological field which is a topological extension of the field of p-adic numbers; this topological field is a missing example in the classification of complete first countable fields given by Mutylin.

Erratum to ``Locally unbounded topological fields with topological nilpotents'' (Fund. Math. 173 (2002), 21–32)

“…There is a completely analogous error at the end of the proof of Lemma 3.3 in [1]. The sentence "We choose k ≥ 2m such that 3 log(n)/n ≤ 1/(2m) for all n ≥ k" should be replaced with the following: We choose k ≥ 2m + 2 such that log(p(n)) + log(N (2)) + 3 log(n)…”

Size functions

“…Condition (N1) is dispensable, in the sense that the topologies induced in [3] by functions satisfying (N1)-(N5) can be induced by functions satisfying only (N2)-(N5). Indeed, in [3] Marcos requires N (0) = 2, except for size functions that satisfy conditions (N3 ) and (N4 ).…”

“…Indeed, in [3] Marcos requires N (0) = 2, except for size functions that satisfy conditions (N3 ) and (N4 ).…”

“…The topology constructed in Section 3 of [3] is described in Theorem 1 below. We have replaced statements in [3] using logarithms with corresponding ones using exponents in order to make the analogy with Theorem 2 more apparent.…”

“…We have replaced statements in [3] using logarithms with corresponding ones using exponents in order to make the analogy with Theorem 2 more apparent.…”