Show that C[0, 1] is a metric space with metric \(d_{\infty}(f,g):=||f-g||_{\infty}\). 

An open ball in a metric space (X, d) is given by

 \(B_d(x,R):=\{ {y \in X:d(y,x)<R }\}\)

Let (X, d) be your favourite metric (X, d). How does open ball in (X, d) look like ?

Visualize the open ball B(f, R) in \(\left(C[0,1],d_{\infty}\right)\), where f is the identity function. We say that Y ⊆ X is open in X if for every y ∈ Y, there exists r > 0 such that B(y, r) ⊆ Y, that is,

{z ∈ X : d(z, y) < r} ⊆ Y.

Give five of your favourite open subsets of \(\mathbb{R^2}\) endowed with any of your favourite metrics.

Give five of your favourite non-open subsets of \(\mathbb{R^2}\) .

Let B[0, 1] denote the set of all bounded functions \(f:[0,1]\rightarrow \mathbb{R}\) endowed with the metric \(d_{\infty}\). Show that C[0, 1] can not be open in B[0, 1].

Show that the open unit ball in \((C[0,1],d_{\infty})\) can not be open in (C[0, 1], \(d_1\)), where 

\(d_1(f,g)=\int_{[0,1]}|f(t)-g(t)|dt\).

Show that the open unit ball in (C[0, 1], \(d_1\)) is open in (C[0, 1], \(d_{\infty}\)).

Consider the first quadrant of the plane with usual metric. Note that the open unit disc there is given by

\(\{ \left( x,y \right) \in \mathbb{R^2}:x \ge 0 ,y\ge 0, x^2+y^2<1\}\).

We say that a sequence {\(x_n\)} in a metric space X with metric d converges to x if \(d\left(x_n,x\right) \rightarrow 0\)   as   \(n \rightarrow \infty\).

Discuss the convergence of \(f_n(t)=t^n\) in (C[0, 1], \(d_1\)) and (C[0, 1], \(d_{\infty}\)).

Every metric space (X, d) is Hausdorff: For distinct x, y ∈ X, there exists r > 0 such that \(B_d(x,r)\cap B_d(y,r)=\emptyset\). In particular, limit of a convergent sequence is unique.