Let $e_i=(0,\dots,0,1,0,\dots,0,\dots)$, where 1 appears in the $i$th place. Let $X$ be the set of all the points $e_i$. Show that $X$ is closed, bounded and non-compact.
It is bounded because for any $x\in X$, $X\subseteq B(x,1)$ and it is not compact because the set $\{B(e_i,1/2):i\in\mathbb{N}\}$ has no finite subcover.
I'm having trouble showing $X$ is closed.
Best Answer
HINT: Let $\langle x_n:n\in\Bbb N\rangle\in\Bbb R^{\infty}\setminus X$. If there is an $n\in\Bbb N$ such that $x_n\notin\{0,1\}$, it’s not hard to find an open nbhd of $x$ that is disjoint from $X$. The only other possibilities are that $x_n=0$ for all $n\in\Bbb N$, or that there are distinct $m,n\in\Bbb N$ such that $x_m=x_n=1$; in each case it is again pretty easy to specify an open nbhd of $x$ that is disjoint from $X$.
By the way, whether $B(x,1)$ contains $X$ depends on what product metric you’re using, so you need to specify the metric.