The Baire category theorem (BCT) asserts that a countable intersection of open dense sets is still dense. A topological space for which the BCT holds is called a Baire space. Examples include locally compact spaces and complete metric spaces. The BCT has major applications in real and functional analysis, such as the open mapping theorem, or the existence of many nowhere differentiable functions.
A meager set is a set which is a countable union nowhere dense sets. They are also called sets of first category. Nonmeager sets are called sets of second category. An equivalent formulation of the BCT is that all open subsets of a topological space $X$ are sets of second category.