Definition:P-adic Topology
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Definition
Let $\Z$ denote the set of integers.
Let $p \in \Z$ be a fixed prime number.
Let $\BB$ be the set of all sets $\map {U_\alpha} n$ defined as:
- $\map {U_\alpha} n = \set {n + \lambda p^\alpha: \lambda \in \Z}$
Then $\BB$ is the basis for a topology $\tau$ on $S$.
$\tau$ is referred to as the $p$-adic topology (on $\Z$).
The topological space $T = \struct {S, \tau}$ is referred to as the $p$-adic (topological) space.
Also see
- Results about the $p$-adic topology can be found here.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (next): Part $\text {II}$: Counterexamples: $59$. The $p$-adic Topology on $Z$