Baire Category Theorem.
Let (X,d) be a complete metric space. Let {An} be a collection of closed subset. Suppose there is an open ball U in the union of An's. Then there is a n s.t U<An
<=> a countable union of closed nowhere dense subset has no interior point.

함수해석에서 나오는 BCT인데요.. 도대체 이게 Category랑 뭔상관인데??하고 줄곧 생각했는데, 우연히 책을보니 써있네요.

Let X be a metric space.
X is said to be of the first category if it's a countable union of closed sets without interior points.
Otherwise, it is said to be of the second category. so the Baire Category Theorem says that every complete metric space is of the second category.
(발번역 ㅈㅅ)  라고 하네요..

그래서 이게 이름이 왜 이래? 하는 Theorem 있으신가요?



시험공부하기 싫으니까 이런게 눈에 다 들어오네..