PlanetMath (more info)
 Math for the people, by the people. Sponsor PlanetMath
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
Owner confidence rating: Very low Entry average rating: No information on entry rating
[parent] balls in ultrametric spaces are clopen subsets (Example)

In an ultrametric space, both open and closed balls are clopen subsets.

It is indeed straightforward (exercise!) to show that the set of all open balls of radius $r$ , centered in any of the points of a closed ball of radius $r$ , forms a partition of the latter.

Thus, in particular, any point of a closed ball is an interior point, and the same holds for the complement of an open ball.




"balls in ultrametric spaces are clopen subsets" is owned by MFH.
(view preamble | get metadata)

View style:


This object's parent.
Log in to rate this entry.
(view current ratings)

Cross-references: complement, interior point, partition, points, radius, clopen subsets, closed balls, open, ultrametric space

This is version 2 of balls in ultrametric spaces are clopen subsets, born on 2008-08-31, modified 2008-08-31.
Object id is 10970, canonical name is BallsInUltrametricSpacesAreClopenSubsets.
Accessed 526 times total.

Classification:
AMS MSC54D05 (General topology :: Fairly general properties :: Connected and locally connected spaces )

Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add example | add (any)