Editing P-adic number (section)

== Topological properties ==
The {{mvar|p}}-adic valuation allows defining an [[absolute value (algebra)|absolute value]] on {{mvar|p}}-adic numbers: the {{mvar|p}}-adic absolute value of a nonzero {{mvar|p}}-adic number {{mvar|x}} is
<math display="block">|x|_p = p^{-v_p(x)},</math>
where <math>v_p(x)</math> is the {{mvar|p}}-adic valuation of {{mvar|x}}. The {{mvar|p}}-adic absolute value of <math>0</math> is <math>|0|_p = 0.</math> This is an absolute value that satisfies the [[strong triangle inequality]] since, for every {{mvar|x}} and {{mvar|y}} one has
* <math>|x|_p = 0</math> if and only if <math>x=0;</math>
* <math>|x|_p\cdot |y|_p = |xy|_p</math>  
* <math>|x+y|_p\le \max(|x|_p,|y|_p) \le |x|_p + |y|_p.</math>

Moreover, if <math>|x|_p \ne |y|_p,</math> one has <math>|x+y|_p = \max(|x|_p,|y|_p).</math>

This makes the {{mvar|p}}-adic numbers a [[metric space]], and even an [[ultrametric space]], with the {{mvar|p}}-adic distance defined by
<math>d_p(x,y)=|x-y|_p.</math>

As a metric space, the {{mvar|p}}-adic numbers form the [[completion (metric space)|completion]] of the rational numbers equipped with the {{mvar|p}}-adic absolute value. This provides another way for defining the {{mvar|p}}-adic numbers. However, the general construction of a completion can be simplified in this case, because the metric is defined by a discrete valuation (in short, one can extract from every [[Cauchy sequence]] a subsequence such that the differences between two consecutive terms have strictly decreasing absolute values; such a subsequence is the sequence of the [[partial sum]]s of a {{mvar|p}}-adic series, and thus a unique normalized {{mvar|p}}-adic series can be associated to every equivalence class of Cauchy sequences; so, for building the completion, it suffices to consider normalized {{mvar|p}}-adic series instead of equivalence classes of Cauchy sequences).

As the metric is defined from a discrete valuation, every [[open ball]] is also [[closed ball|closed]]. More precisely, the open ball <math>B_r(x) =\{y\mid d_p(x,y)<r\}</math> equals the closed ball <math>B_{p^{-v}}[x] =\{y\mid d_p(x,y)\le p^{-v}\},</math>  where {{mvar|v}} is the least integer such that <math>p^{-v}< r.</math> Similarly, <math>B_r[x] = B_{p^{-w}}(x),</math> where {{mvar|w}} is the greatest integer such that <math>p^{-w}>r.</math>

This implies that the {{mvar|p}}-adic numbers form a [[locally compact space]] ([[locally compact field]]), and the {{mvar|p}}-adic integers—that is, the ball <math>B_1[0]=B_p(0)</math>—form a [[compact space]].