Example of closed unit ball?

Question

I am not understanding the concept of ball on a set $E$ and closed unit ball $B_1$ in $B(E)$.
I need to prove or disprove by example that if the closed unit ball $B_1$ is compact or not in a metric space $X$.
Can you give me an any example of ball and closed unit ball to help me to understand the concept itself?
Also how can I connect the concept of unit ball to a finiteness of a set $E$?

Answer 1

Given a metric space $E$, the closed ball about $x$ of radius $r$ is the set $\{y\in E\ :\ d(x, y) \leq r \}.$ An important result here is the Bolzano-Weirerstrass theorem, or sequential compactness theorem.

http://en.wikipedia.org/wiki/Bolzano%E2%80%93Weierstrass_theorem#Sequential_compactness_in_Euclidean_spaces

Use this result to work out where you should try looking for examples of non compact unit balls.