The definition of an open set in a Euclidean space $\mathbb{R}^n$ as stated in Wikipedia is:
a subset $U \subset \mathbb{R}^n$ is open if every point in $U$ is the center of an open ball contained in $U$
What I want to know is why specifically an open ball is required. Why not a closed ball? Because the way I see it – if you can show that a point in the set is the center of an open ball, you could always define a smaller closed ball around it. So why not take closed balls instead?
Also, from here: Can an open ball have just one point. As per my understanding it cannot. Please clarify. , I understand that open balls containing just one point are also possible. So, possibly, even for a closed set, we could define points on the boundary as being the center of an open ball. So (to avoid this), should the definition also state explicitly that the open balls should have more than one point. OR is the structure of the Euclidean space such that open balls containing one point do not exist.
Best Answer
If you allow closed balls, then you also allow singleton sets: a closed ball of radius zero around a point $x\in\mathbb{R}^{n}$ looks like $$\{y\in\mathbb{R}^{n}:\lVert y-x\rVert\leq0\} = \{x\}.$$ We don't want singletons to be open sets (if all singletons are open, then every set is open, because a union of open sets should be open as well). It's cleaner to just focus on open balls rather than have to qualify which closed balls are allowed.