Show that \(X=A(0,r,R)\) (open annulus) and \(Y= \mathbb{T}\)(unit circle) are not homeomorphic

Let \(X=S^n \backslash \{(0,\ldots ,1)\} \subseteq \mathbb{R^{n+1}}\) and \(Y= \mathbb{R^n}\) , and define

\(f:X \rightarrow Y\) and \(g:Y \rightarrow X\) by

\(f(x_1, \ldots ,x_{n+1}) =\large( \frac{x_1}{1-x_{n+1}}, \ldots , \frac{x_n}{1-x_{n+1}}),\\g(x_1, \ldots ,x_n)=\large(\frac{2x_1}{{||x||_2}^2+1} ,\ldots ,\frac{2x_n}{{||x||_2}^2+1} , \frac{{||x||_2}^2-1}{{||x||_2}^2+1} )\)

Verify that \(f\) and \(g\) are homeomorphisms such that \(f^{-1}=g\).

Show that the graph \(G\) of a continuous function \(f:X \rightarrow Y\) (with the subspace topology inherited from \(X \times Y\) ) is homeomorphic to \(X\).

Consider the continuous function \(f: \mathbb{R} \backslash \{0\} \rightarrow \mathbb{R}\) by \(f(x)=1/x\). The graph of f is the hyperbola \(xy=1 \). One may conclude from the last exercise that \(\mathbb{R} \backslash \{0\}\) is homeomorphic to \(xy=1 \).

Consider the metric spaces \((X_i,d_i),i=1 ,\cdots ,m\). Check that \(d(x,y)=max_id_i(x_i,y_i)\) defined on \(X= {\Pi ^m}_{i=1} X_i\) is a metric. The metric topology coincides with the product topology on \(X\). This may be concluded from the fact that for \(x=(x_1, \cdots ,x_m),\mathbb{B_d}(x,r) ={\Pi ^m}_{i=1} \mathbb{B_{d_i}}(x_i,r)\).

Show that projections \(\pi_j:{\Pi^m}_{i=1}X_i \rightarrow X_j\) is continuous.

Verify that \(\mathbb{B_b}\) and \(\mathbb{B_p}\) are basis for a topology on \({\Pi^m}_\alpha X_\alpha\):

(1)  \(\mathbb{B_b}=\{\Pi_{\alpha \in I } U_\alpha :U_\alpha \; is \; open \; in X_\alpha \}\) .

(2) \(\mathbb{B_b}=\{\Pi_{\alpha \in I } U_\alpha \in \mathbb{B_b}:U_\alpha=X_\alpha\) for finitely many values of \(\alpha\}\).

Suppose that each \(X_\alpha\) contains a non-empty proper open subset \(U_\alpha\). Show that \(\Omega_\mathbb{B_p} = \Omega_\mathbb{B_b}\) iff \(I\) has finite cardinality.

Show that the product topology is the smallest topology which makes all projections \(\pi_\alpha\) continuous.

Let \(f:A \rightarrow \Pi_\alpha X_\alpha\) be given by \(f(a)=(f_\alpha(a))\), where the functions \(f_\alpha : A \rightarrow X_\alpha\) are given. Suppose \(\Pi_\alpha X_\alpha\) has either box topology or product topology. Show that if f is continuous then each \(f_\alpha\) is continuous.

Show that \(D : \mathbb{R^ \omega} \times \mathbb{R^ \omega} \rightarrow \mathbb{R}\) given by

\(D((x_n),(y_n)):=sup_n \frac{min \{|x_n-y_n|,1\}}{n}\)

defines a metric on \(\mathbb{R^ \omega}\) . Verify further that \(D \le d_u\) , where \(d_u\) is the uniform metric.