Baire category

Baire category is a concept in mathematics that classifies sets based on their size and "thickness" within a space. In simple terms, a set can be considered "large" if it's dense and covers many parts of the space, or "small" if it's nowhere dense or thinly spread out. Baire's theorem states that in certain spaces, such as complete metric spaces, the union of countably many "small" sets is still "small." This helps mathematicians understand which parts of a space are "typical" or "rare," and it’s fundamental in analyzing the structure of functions and spaces in analysis.