: Ultrametric

10.13 Ultrametric

[UNACCESSIBLE UUID ’0WK’]

Definition 302

[0WM] An ultrametric space is a metric space in which the triangle inequality is reinforced by the condition

\begin{equation} d(x,y)≤\max \{ d(x,z),d(z,y)\} ~ ~ .\label{eq:ultrametrico} \end{equation}

303

E303

[0WN]Show that 303 implies that \(d\) satisfies the triangle inequality.

E303

[0WP]Note that if \(d(x,y)≠ d(y,z)\) then \(d(x,z)= \max \{ d(x,y),d(y,z)\} \). Hidden solution: [UNACCESSIBLE UUID ’0WQ’] Intuitively, all triangles are isosceles, and the base is shorter than equal sides.

E303

[0WR] Consider two balls \(B(x,r)\) and \(B(y,𝜌)\) radius \(0{\lt}r≤ 𝜌\) that have non-empty intersection: then \(B(x,r)⊆ B(y,𝜌)\).

Similarly for the disks \(D(x,r){\stackrel{.}{=}}\{ y∈ X : d(x,y)≤ r\} \) and \(D(y,r)\).

Hidden solution: [UNACCESSIBLE UUID ’0WS’]

E303

[0WT]Show that two balls \(B(x,r)\) and \(B(y,r)\) of equal radius are disjoint or are coincident; in particular they are coincident if and only if \(y∈ B(x,r)\). Similarly for the discs \(D(x,r){\stackrel{.}{=}}\{ y∈ X : d(x,y)≤ r\} \) and \(D(y,r)\).

Hidden solution: [UNACCESSIBLE UUID ’0WV’]

E303

[0WW] Show that every open ball \(B(x,r)\) is also closed. Show that every disk \(D(x,r)\) with \(r{\gt}0\) is also open. Hidden solution: [UNACCESSIBLE UUID ’0WX’] By the exercise 1, there follows that the space is totally disconnected.

E303

[0WY] Let \(𝜑:[0,∞)→ [0,∞)\) be a function that is continuous in zero, monotonically weakly increasing and with \(𝜑(x)=0\iff x=0\). Show that \(\tilde d=𝜑◦ d\) is still an ultrametric. Show that spaces \((X,d)\) \((X,\tilde d)\) have the same topology.

Compare with the exercise 5, notice that we do not require \(𝜑\) to be subadditive.

[UNACCESSIBLE UUID ’0WZ’]

Ultrametric space of sequences

Let’s build this example of ultrametric on the space of sequences.

Definition 304

[0X0] Let \(I\) be a non-empty set, with at least two elements. Let \(X=\{ f:ℕ→ I\} =I^ℕ\) be the space of sequences. Let \(x,y∈ X\). If \(x=y\) then we set \(d(x,y)=0\). ¹ If \(x≠ y\), we set

\begin{equation} c(x,y)=\min \{ n≥ 0, x(n)≠ y(n)\} \label{eq:c_ ultrametrica_ succ} \end{equation}

305

to be the first index where the sequences are different; then we define \(d(x,y)=2^{-c(x,y)}\).

Remark 306

[0X1] Because of the exercise 6, we could equivalently define \(d(x,y)=\varepsilon _{c(x,y)}\) with \(\varepsilon _ n{\gt}0\) infinitesimal decreasing sequence.

E306

[0X2]Prerequisites:304.Show that \(d(x,y)≤ \max \{ d(x,z),d(y,z)\} \).

Hidden solution: [UNACCESSIBLE UUID ’0X3’]

E306

[0X4]Topics:complete. Prerequisites:304.Show that \((X,d)\) is complete. Hidden solution: [UNACCESSIBLE UUID ’0X5’]

E306

[0X6]Topics:compact.

Prerequisites:304.

Show that \((X,d)\) is compact if and only if \(I\) is a finite set.

Hidden solution: [UNACCESSIBLE UUID ’0X7’]

E306

[0X8]Prerequisites:304,2. Suppose that \(I\) is a group; then \(X\) is a group (it is the Cartesian product of groups); and multiplication is carried out ”component by component”. Show that the product in \(X\) is a continuous operation, and so for the inversion map. So \((X,d)\) is a topological group.

Hidden solution: [UNACCESSIBLE UUID ’0X9’] [UNACCESSIBLE UUID ’0XB’]

[0XC]Prerequisites:304,2. Let \(I\) be a set of cardinality 2, then the space \((X,d)\) is homeomorphic to the Cantor set (with the usual Euclidean metric \(|x-y|\)).

Hidden solution: [UNACCESSIBLE UUID ’0XD’]

Combining this result with 4 we get that the Cantor set (with its usual topology) can be endowed with an abelian group structure, where the sum and inverse are continuous functions; This makes it a topological group.