Real Analysis Open sets & Open Balls I very well know that every open ball is an open set. and that every open set need not be an open ball. But illustrate me some counter example.
 A: Here is a counterexample.
Consider the set $\Bbb R$ of real numbers.  Let $B(0, 2)$ represent the ball of radius $2$ around $0$, i.e, the interval $(0-2, 0+2) = (-2,2)$.
Now let $B(5,2)$ represent the ball of radius $2$ around $5$, i.e., the interval $(5-2, 5+2) = (3,7)$.
The union of these two balls, $B(0,2) \cup B(5,2)$, is open, but it is not itself a ball because every open ball in $\Bbb R$ can be expressed as an open interval, but how do you express that union as a single open interval?  If it helps, feel free to draw a picture of the real line $\Bbb R$ and the intervals I mentioned.
A: The most simple examples of open sets, which are not balls, in every metric space are $\emptyset$ and the space itself, which are open.
Even if you consider those sets to be balls with radius $0$ or $\infty$, respectively, you can take the union of two or more open balls, which is not necessarily an open ball anymore (see user46944's answer for the one-dimensional case).
What you can actually show instead is that every open set in a metric space can be written as a union of open balls (see this quesion). In $\mathbb{R}^n$ you can furthermore show that a set is open if and only if it is a countable union of open balls (see this question).
