Bounded and Closed but not Compact I had been proposed to construct not compact set that is however bounded and closed.
I could easily imagine from the different metric - such as discrete metric where 
$d(x,y) =0$ if $x=y$ and $d(x,y) =1$ if $x \neq y$ then M itself is closed since it is metric itself and bounded by $D(x,2)$, but is not compact since there's $D(x,1/2)$ which is open cover of $M$ with no finite sub-cover 
any how to construct bounded and closed but not compact set in complete metric?

If $(X,||*||)$ is a normed space and if $B=\{x \in X: ||x|| \le 1\}$, then $B$ is closed and bounded. The following assertions are equivalent:
$(1)$: $B$ is compact;
$(2)$: $ \dim X < \infty$.
