All Questions page - 59

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.

(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 \(\Omega\).

(Co-countable Topology) For a set \(X\), define \(\Omega\) to be the collection of subsets \(U\) of \(X\) such that either \(U=\emptyset\) or \(X \backslash U\) is countable. Show that \(\Omega\) is a topology on \(X\).

Let \(\Omega\) be the collection of subsets \(U\) of \(X:=\mathbb{R}\) such that either \(X \backslash U= \emptyset\) or \(X\backslash U\) is infinite. Show that \(\Omega\) is not a topology on \(X\).