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.

Show that any finite set is compact.

Let \(K \subseteq Y \subseteq X\). Show that \(K\) is compact in \(X\) iff \(K\) is compact in \(Y\).

We invoke a classical result from Analysis, which characterizes all compact subsets of \(\mathbb{R^n}\) .

Let p be a polynomial in the real variables \(x,y\). Then the zero set \(Z(p)\) of \(p\) is always closed. However, \(Z(p)\) may or may not be compact. For instance, if \(p(x,y)=x^2=y^2-1\)then \(Z(p)\) being the unit circle is compact. If \(p(x,y)=x\) then \(Z(p)\) is the Y -axis, which is certainly non-compact.