A fundamental concept in topology and real analysis
Let A be a subset of ℝ. An open covering of A is a collection of open sets {Uᵢ}ᵢ∈I such that:
A ⊂ ⋃i∈I Uᵢ
Let's start with a set S (the thick blue line segment). This is the set we want to 'cover'.
Each set Uᵢ is open (contains no boundary points)
Their union contains the entire set S
The collection {Uᵢ} can have infinitely many sets
Think of an open covering like putting overlapping bubbles over every point of your set. Each bubble is an open set, and together they must cover everything. The bubbles can overlap (and usually do!), and you might need infinitely many of them—unless your set is compact (closed and bounded), in which case finitely many bubbles always suffice!