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.

Let \(p\) be a non-constant polynomial in the complex variables \(z_1, \cdots ,z_m\). Show that the zero set \(Z(p)\) of \(p\) is compact iff \(m=1\).

Show that every subset of \(\mathbb{R}\) with co-finite topology is compact.

Show that the continuous image of a compact space is again compact.

Let \(X:=\{(u,v) \in \mathbb{R^2}:- \pi \le u \le \pi ,-1 \le v \le1\}\) (Rectangle) and let \(Y:=\{(x,y,z) \in \mathbb{R^3}:x^2+y^2=1,|z| \le1\}\) (Cylinder). Verify the following:

(1) If \(u-u^{\prime} =\pm2 \pi\)then define \((u,v) \sim(u^\prime ,v^\prime)\) iff \(v=v^\prime\) . Otherwise, define \((u,v) \sim(u^\prime ,v^\prime)\) iff \((u,v) = (u^\prime ,v^\prime)\). Then \(\sim\) defines an equivalence relation on \(X\).

(2) \(f:X \rightarrow Y,f(u,v)=(cosu,sinu,v)\) is a continuous surjection. Conclude that \(X / \sim\) and \(Y\) are homeomorphic.

Let \(X:=[0,1] \times[0,1]\) (Unit Square) and let \(Y= \mathbb{T} \times \mathbb{T}\) (Torus). Verify the following:

(1) If \(s_1-s_2= \pm1\) then define \((s_1,t_1) \sim (s_2,t_2)\) iff \(t_1=t_2\). If \(t_1-t_2 = \pm1\) then define \((s_1,t_1) \sim (s_2,t_2)\) iff \(s_1=s_2\). If \((s_1,t_1),(s_2,t_2) \in [0,1] \times [0,1]\) then \((s_1,t_1) \sim (s_2,t_2)\) provided \(s_1=s_2\) and \(t_1=t_2\). Then \(\sim\) defines an equivalence relation on \(X\).

(2) \(f:X \rightarrow Y\) given by \(f(s,t)=(e^{2\pi is},e^{2\pi it})\) is a continuous surjection. Conclude that \(X / \sim\) and \(Y\) are homeomorphic.