Borel sets are a collection of sets that can be formed from open sets through countable unions, countable intersections, and relative complements. They are fundamental in the field of measure theory and topology, providing a way to define measurable spaces. The Borel σ-algebra, which consists of all Borel sets, is generated from the open sets in a given topological space. This structure allows for the rigorous treatment of concepts like continuity and convergence in mathematical analysis, making Borel sets essential for understanding real-valued functions and probability theory.