PlanetMath: ultrametric space

(more info)

Math for the people, by the people.

Main Menu

ultrametric space

(Definition)

The metric space $(X,\,d)$ is called an ultrametric space, if its metric is an ultrametric, i.e. if

$\displaystyle d(x,\,z) \leqq \max \{d(x,\,y),\,d(y,\,z)\} \quad \forall x,\,y,\,z\in X.$

Example. The field $\mathbb{Q}$ together with any of its -adic metrics

$\displaystyle d_p(x,\,y) = \vert x-y\vert _p,$

where $\vert\cdot\vert _p$ is the

-adic valuation of $\mathbb{Q}$ , forms an ultrametric space.

"ultrametric space" is owned by pahio.

(view preamble)

Log in to rate this entry.
(view current ratings)

Cross-references: field, ultrametric, metric, metric space

This is version 3 of ultrametric space, born on 2005-01-03, modified 2005-01-27.
Object id is 6612, canonical name is UltrametricSpace.
Accessed 1558 times total.

Classification:

54E35 (General topology :: Spaces with richer structures :: Metric spaces, metrizability)

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact