Baire's theorem

Baire’s theorem states that in a complete, well-behaved space (called a Banach space), the intersection of countably many dense sets is also dense. This means if each of these sets is "spread out" throughout the entire space, then they overlap in a way that still covers densely somewhere. In practical terms, it ensures that “large,” widespread properties are preserved even after many such sets are intersected. It is foundational in analysis, helping to understand the structure of function spaces and ensuring certain “generic” properties exist within complex mathematical contexts.