Unit neighborhoods in topological rings

Abstract

The concepts of open unit ball and closed unit ball in a real or complex normed space are naturally extended to the scope of topological rings with unity. We then define a type of open (closed) sets called open (closed) unit neighborhoods of $0$. We show among other things that in $\mathbb{R} $ and $\mathbb{C} $ the only non-trivial open and closed unit neighborhoods of $0$ are the open unit ball and the closed unit ball, respectively.