Definition:Closed Ball/P-adic Numbers
< Definition:Closed Ball(Redirected from Definition:Closed Ball in P-adic Numbers)
Jump to navigation
Jump to search
Definition
Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $a \in \Q_p$.
Let $\epsilon \in \R_{>0}$ be a strictly positive real number.
The closed $\epsilon$-ball of $a$ in $\struct {\Q_p, \norm {\,\cdot\,}_p }$ is defined as:
- $\map { {B_\epsilon}^-} a = \set {x \in R: \norm {x - a}_p \le \epsilon}$
Radius
In $\map { {B_\epsilon}^-} a$, the value $\epsilon$ is referred to as the radius of the closed $\epsilon$-ball.
Center
In $\map { {B_\epsilon}^-} a$, the value $a$ is referred to as the center of the closed $\epsilon$-ball.
Also see
- Definition:Open Ball in P-adic Numbers
- Definition:Sphere in P-adic Numbers
- P-adic Closed Ball is Instance of Closed Ball of a Norm
Sources
- 2007: Svetlana Katok: p-adic Analysis Compared with Real: $\S 2.1$ Elementary topological properties