Let \(X\) be a set and let \(d:X\times X\to \mathbb R\) be defined as
\[d\left( x,y\right) = \begin{cases}
0,& x=y,\\
1,& x\not=y.
\end{cases}\] Show that \(d\) is a metric on \(X\) (which we call the discrete
metric).
Show in addition that if \(X\) is a
non-trivial vector space, the discrete metric can’t be induced by a
norm. In other words, show that there exists no norm on \(X\), \(\left\lVert\cdot\right\rVert\), such that
\[d\left( x,y\right)=\left\lVert
x-y\right\rVert.\]
(Assignment sheet 2 problem 3) In the space \(C([a,b])\) of continuous functions defined on a closed interval \([a,b]\) (for \(a<b\)), let \[d_1(f,g): = \int_a^b|f(t)-g(t)| \, dt.\] Show that \(d_1\) is a metric on \(C([a,b])\).
(Assignment sheet 2 problem 7)
Show that in any metric space \((X,d)\) the set \(\{x\}\), consisting of a single point \(x \in X\), is closed.
Show that in any metric space \((X, d)\) the closed ball \(\overline{B}_r(x):= \{y \in X: d(y,x) \leq r \}\), of radius \(r>0\) centred at \(x \in X\), is closed.
(Assignment sheet 2 problem 13) Give an example of a metric space \(X\) and an \(x \in X\) such that \(\overline{B}_1(x) \not= \overline{B_1(x)};\) that is, the closure of the open ball is not necessarily the closed ball!!