Can an open ball have just one point. As per my understanding it cannot. Please clarify. I am new to functional analysis and am just learning. To my understanding an open ball must have at least 2 points else its definition will not be satisfied.
Now if I have just an empty set and this open ball, why cannot it constitute a Topology.
I see it satisfying intersection and union conditions. 
I agree that the question is too basic!
Definition of open ball : $$B_r(x)= \{y \in E | d(x,y)<r \}$$
$(E,d)$ is the metric space.

It depends on the space. In $\mathbb{N}$ with the usual topology inherited from the real line, the open 1-ball about any point is just that point. Such points are said to be isolated.
Your example is correct. It is a trivial space consisting of a single point, and you have observed the only topology that exists for that space.
