The Baird's theorem on continuous images

Baire's theorem states that in a complete metric space (a well-behaved mathematical space), the intersection of countably many dense open sets is dense. When considering continuous functions, a key implication is that the image of a complete metric space (or an open subset) under such a function cannot be "small" or "negligible" in a topological sense; it must be "large" or dense in at least some part of the target space. In simple terms, continuous maps from well-structured spaces preserve certain richness or "density," ensuring that the image isn't just a tiny or insignificant subset.