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.

(Co-finite Topology) We declare that a subset $U$ of $\mathbb{R}$ is open iff either $U=\emptyset$ or $\mathbb{R} \backslash U$is finite. Show that R with this “topology” is not Hausdorff.

A subset $U$ of a metric space X is closed if the complement X \$U$ is open. By a neighbourhood of a point, we mean an open set containing that point. A point x ∈ X is a limit point of $U$if every non-empty neighbourhood of x contains a point of U. (This definition differs from that given in Munkres). The set $\overline{U}$ is the collection of all limit points of $U$.

What are the limit points of bidisc in $\mathbb{C^2}$ ?

 Let (X, d) be a metric space and let $U$ be a subset of X. Show that $x\in \overline{U}$, iff for every $x\in \overline{U}$, there exists a convergent sequence $\{x_n \} \subseteq U$ such that $Lim_{n \rightarrow\infty}x_n=x$.

 Describe all topologies on a 2-point set. Give five topologies on a 3-point set

Let $(X,\Omega)$ be a topological space and let $U$ be a subset of $X$. Suppose for every $x\in U$ there exists $U_x\in \Omega$ such that $x \in U_x \subseteq \Omega$. Show that $U$ belongs to $\Omega$.