Baire Category Theorem

The Baire Category Theorem states that in complete metric spaces (like the real numbers), the intersection of countably many dense open sets is still dense. In simpler terms, if you have a space where you can get arbitrarily close to any point, combining infinitely many "thick" open sets will still fill the space in some way. This theorem has profound implications in analysis and topology, showing that certain mathematical objects cannot be "small" or "sparse," and it helps distinguish between different types of size or density in mathematical spaces.