Editing Ultrametric space (section)

== Properties ==
[[File:Strong_triangle_ineq.svg|thumb|upright=1.25|In the triangle on the right, the two bottom points ''x'' and ''y'' violate the condition ''d''(''x'', ''y'') ≤ max{''d''(''x'', ''z''), ''d''(''y'', ''z'')}.]]
From the above definition, one can conclude several typical properties of ultrametrics. For example, for all <math>x,y,z \in M</math>, at least one of the three equalities <math>d(x,y) = d(y,z)</math> or <math>d(x,z) = d(y,z)</math> or <math>d(x,y) = d(z,x)</math> holds. That is, every triple of points in the space forms an [[isosceles triangle]], so the whole space is an [[isosceles set]].

Defining the [[open ball|(open) ball]] of radius <math>r > 0</math> centred at <math>x \in M</math> as <math>B(x;r) := \{y \in M \mid d(x,y) < r\}</math>, we have the following properties:
* Every point inside a ball is its center, i.e. if <math>d(x,y)<r</math> then <math>B(x;r)=B(y;r)</math>.
* Intersecting balls are contained in each other, i.e. if <math>B(x;r)\cap B(y;s)</math> is [[Empty set|non-empty]] then either <math>B(x;r) \subseteq B(y;s)</math> or <math>B(y;s) \subseteq B(x;r)</math>.
* All balls of strictly positive radius are both [[Open set|open]] and [[closed set]]s in the induced [[topology]]. That is, open balls are also closed, and closed balls (replace <math><</math> with <math>\leq</math>) are also open.
* The set of all open balls with radius <math>r</math> and center in a closed ball of radius <math>r>0</math> forms a [[partition of a set|partition]] of the latter, and the mutual distance of two distinct open balls is (greater or) equal to <math>r</math>.

Proving these statements is an instructive exercise.<ref>{{cite web |work=Stack Exchange |url=https://math.stackexchange.com/q/1141731 |title=Ultrametric Triangle Inequality }}</ref> All directly derive from the ultrametric triangle inequality. Note that, by the second statement, a ball may have several center points that have non-zero distance. The intuition behind such seemingly strange effects is that, due to the strong triangle inequality, distances in ultrametrics do not add up.