10.15 Circle[2CF]

It is a closed set in \(ℝ^ 2\), so we can think of it as a complete metric space with the Euclidean distance \(d(x,y)=|x-y|_{ℝ^ 2}\).

As usual, given \(t∈ℝ\), we indicate with \([t]\) the class of elements in \(ℝ/2𝜋\) equivalent to \(t\).

E314

[0Y5]Consider the map

\begin{eqnarray*} Φ : ℝ/2𝜋 & →& S^ 1\\ {} [t] & ↦ & (\cos (t),\sin (t)) \end{eqnarray*}

Show that it is well-defined and bijective.

Hidden solution: [UNACCESSIBLE UUID ’0Y6’]

E314

[0Y7]Through this bijection we transport the Euclidean distance from \(S^ 1\) to \(ℝ/2𝜋\) defining

\[ d_ e([s],[t])=|Φ([s])-Φ([t])|_{ℝ^ 2} ~ ~ . \]

With this choice the map \(Φ\) turns out to be an isometry between \((S^ 1,d)\) and \((ℝ/2𝜋,d_ e)\) (see the Definition 297). So the latter is a complete metric space.

With some simple calculations it can be deduced that

\[ d_ e([s],[t])= \sqrt{ |\cos (t)-\cos (s)|^ 2 + |\sin (t)-\sin (s)|^ 2}= \sqrt{ 2 - 2 \cos (t-s)} ~ ~ . \]

Then we define the function

\[ d_ a([s],[t]) = \inf \{ |s-t-2𝜋 k| : k∈ℤ\} ~ ~ , \]

show that it is a distance on \(ℝ/2𝜋\).

Hidden solution: [UNACCESSIBLE UUID ’0Y8’]

E314

[0Y9]Show that \(d_ a([s],[t])\) is the length of the shortest arc in \(S^ 1\) that connects \(Φ([s])\) to \(Φ([t])\).

E314

[0YB]Show that distances \(d_ a\) and \(d_ e\) are equivalent, proving that \(\frac{2}{𝜋}d_ a≤ d_ e≤ d_ a\).

Hidden solution: [UNACCESSIBLE UUID ’0YC’]

E314

[0YD] Prerequisites:1.One can easily show that a function \(f:ℝ/2𝜋→ X\) can be seen as a periodic function \(\tilde f:ℝ→ X\) of period \(2𝜋\), and vice versa.

This can be easily obtained from the relation \(f([t])=\tilde f(t)\) where \(t\) is a generic element of its equivalence class \([t]\). Assuming that \(\tilde f\) is periodic (with period \(2𝜋\)), the above relation allows to derive \(f\) from \(\tilde f\) and vice versa.

Show that \(f\) is continuous if and only if \(\tilde f\) is continuous.

E314

[0YF] Prerequisites:3.Let \((X,𝜏)\) be the compactified line, the topological space defined in 3. Show that it is homeomorphic to \(S^ 1\).

[UNACCESSIBLE UUID ’0YG’]