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.

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.