Open Set

Open sets can be formalized with various degrees of generality[1].
In general topological spaces,
let X be a set, and τ be a family of sets. τ is a topology on X if:
- X and ∅ are in τ: X∈τ,and ∅∈τ
- any union of sets in τ is in τ
- any finite intersection of sets in τ is in τ

the sets in τ are open sets. It means, the openness property of a set is preserved through the process of union and finite intersection.

From the point of view of topology, the open sets can be almost anything. Geometrically, open sets have a little bit of space around each point[2].
Closed sets contain all their boundary points, open sets contain none of theirs. Open sets have "enough space" that there is a ball around every point.
Closed sets can be so sparse they contain no metric balls at all.
Differentiation is usually defined only for function defined on an open set.

[2] "Intuitively, closed set contains points such that if you are on one of them there is some direction such that if you move towards this direction by a so small movement (so small as you want) you Will fall down outside the set. However, for an open set, if you are on any of its points and you take any direction you can always move by a so small movement such you remains inside the set during the movement as well as on the destination point!"

[1]https://en.wikipedia.org/wiki/Open_set
[2]https://math.stackexchange.com/questions/1295138/what-is-the-mathematical-distinction-between-closed-and-open-sets