...1 . A set ACX is called a _/«rs? category set if A is a countable union of nowhere dense sets. 2. A set that is not of the first category is called a set of the second category. 3. The complement of a first-category set is called a residual set. For complete metric spaces, first-category...

...space. 1. A set ACX is called a first category set if ^4 is a countable union of nowhere dense sets. 2. A set that is not of the first category is called a set of the second category. 3. The complement of a first-category set is called a residual set. For complete metric spaces, first-category...

...dense sets might be dense. A set £ С X is said to be meager, or of the first category, if it is the countable union of nowhere dense sets. A set that is not of first category is said to be of the second category. The complement of a set of first category is called...

...r) containing no points of the set E. A set E is of first category in X if it can be represented as a countable union of nowhere dense sets. A set that is not of first category is of second category in X. Show that a complete metric space is a set of second category...

...first category if E is a countable union of non-dense sets in X. A subset E of X which is not a set of the first category is called a set of the second category. Theorem 15.47. (Baire Category Theorem) Let (X, d) be a complete metric space. Then X is a set of the...

...category theorem because of Baire's definition. Definition: Category A set is of first category if it is a countable union of nowhere dense sets. A set that is not of first category is said to be of second category. Thus, the Baire category theorem is more succinctly...