BallsInUltrametricSpacesAreClopenSubsets 2008-08-31 16:36:26 2008-08-31 23:10:54 Example clopen subset 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. % Indeed, if $x$ is any point of a closed ball of radius $r$, % then the open ball of radius $r$, centered in $x$, is contained in the closed ball. % PROOF: % Let x be in the closed ball with center c and radius r. % Let y be in the open ball with center x and radius r. % Then |cy| <= max(|cx|,|xy|) = |cx| = r, i.e. y is in the closed ball. % Thus, the whole open B(x,r) is in the closed ball B'(c,r) % Let |xy| >= r, assume there is z in B(x,r) and B(y,r). % Then |xz|<r,|yz|<r, impossible since r <= |xy| <= max(|xz|,|yz|) < r. % Thus, open balls with centers at distance >= r do not intersect. % This shows that each point of the complement of the open ball % is an interior point of that complement (which in fact contains % the whole open ball of radius r around the given point). % Assume z is in B(x,r) and in B(y,r). % Then |xy| <= max(xz,yz) < r. Thus x is in B(y,r) and y is in B(x,r). % Now any other point t which is in B(x,r) verifies yt <= max(yx,tx) < r % and is in B(y,r). So B(x,r) c B(y,r) and reciprocally, i.e. equality. % Thus 2 open balls of radius r either are the same (if their centers % are at distance < r), or are disjoint (if the centers are at dist. >= r).