18.786 Topics in Algebraic Number Theory, Lecture 7

Remark. The non-archimedean topology: Recall that if K is a field with a val­ uation | |, then it also is a metric space with d(x, y) = |x − y|. The topology has a basis of open neighborhoods given by B(x, ǫ) = {y ∈ K | |x − y| < ǫ}. If the valuation is nonarchimedean, then this metric space or topology is rather bizarre. For instance, the open balls don’t have a unique center: in fact, if we take any y ∈ B(x, ǫ), then y is a center of the ball as well, i.e. B(x, ǫ) = B(y, ǫ)! To see this, use the strong triangle inequality