$\mathbb{R^n}$ is homeomorphic to $\mathbb{R}$ iff $n=1$.

Let $f: A \rightarrow \Pi_{i=1}^m X_i$ be given by $f(a)=(f_1(a), \cdots , f_m(a))$, where the functions $f_i :A \rightarrow X_i (i=1,\cdots ,m)$ are given. Then $f$ is continuous iff each $f_i$ is 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 product topology. Show that $f$ is continuous iff each $f_\alpha$ is continuous.

Let $\{x_n=(x_{n1},x_{n2}, \cdots,) \}$be a sequence in the product space $\Pi_{\alpha \in I} X_\alpha$ with product topology. Then the sequence $\{x_n\}$ converges to$x+(x_1,x_2, \cdots,)\in \Pi_\alpha X_\alpha$ iff for every positive integer $m, \{\pi_m(x_n)=x_{nm} \}$ converges to $x_m$.

$\Omega_\mathbb{B_p} \subseteq \Omega_\mathbb{B_u} \subseteq \Omega_\mathbb{B_b}$ with strict inclusions if $I$ is infinite.

Let $X$ and $Y$ be compact spaces. Assume further that $Y$ is Hausdorff. Let $f:X \rightarrow Y$ be a continuous surjection. Define the equivalence relation $\sim$ on $X$ by $x_1 \sim x_2$ iff $f(x_1)=f(x_2)$. Then $g:X / \sim \rightarrow Y$ given by $g(|x|)=f(x)$ is a well-defined homeomorphism.

Show that in the ring $\mathbb{Z}[\sqrt{-3}]$ the gcd of $4$ and $2+2\sqrt{-3}$ does not exist.

Describe the field of quotients of  $\mathbb{Z} [\sqrt{n}]$ , where $n$ is a square-free integer.

Show that the domains $\mathbb{Z} [\sqrt{-6}]$ and $\mathbb{Z} [\sqrt{-7}]$ are not UFDs.

Let $R$ be an integral domain that is not a field; show that the polynomial ring $R[x]$ is not a PID.

Show that the polynomial ring $F[x,y]$ in two variables over a field $F$ is a UFD but not a PID.