can someone please let me know if the following is correct:
1) Let $\mathbb{Z}$ be the integers endowed with the discrete topology and $\mathbb{N}$ the natural numbers. Is $\mathbb{Z}^{\mathbb{N}}$ a discrete space with the product topoogy?
2) Does $\mathbb{Z}^{\mathbb{N}}$ contain a compact infinite set?
3) Is $\mathbb{Z}^{\mathbb{N}}$ metrizable?
My work:
1) I think this is false, let $A= \{0\} \times \{0\} \times ...$ Suppose $A$ is open in $\mathbb{Z}^{\mathbb{N}}$ then we can find a basic open set $U=\prod_{n \in \mathbb{N}} U_{n}$ such that $(0,0,0,...) \in U \subset A$. By definition of product topology there exists a natural number $J$ such that if $n>J$ then $U_{n} = \mathbb{Z}$. This in turn implies that:
$U_{1} \times U_{2} ...\times U_{J} \times \mathbb{Z} \times \mathbb{Z} ... \subset \{0\} \times \{0\} \times ...$
which is not true since we can pick $z \in \mathbb{Z} \setminus \{0\}$ then $(0,0,...0,z,z,z...)$ is the LHS while not in the RHS.
2) Can we simply say, take $\{0,1\}$ endowed with the discrete topolgy then $\{0,1\}$ is compact since it is finite. But then by Tychonoff theorem $\{0,1\}^{\mathbb{N}}$ is compact and clearly infinite.
3) I think this one is true right? $\mathbb{Z}$ is metrizable (e.g discrete metric) and the countable product of metrizable spaces is metrizable.