Complete the proof of the theorem. This result is called the Baire Category Theorem because it creates two categories of size for subsets in a metric space. A set of “first category” is one that can be written as a countable union of nowhere-dense sets. These are the small, intuitively thin subsets of a metric space. We now see that if our metric space is complete, then it is necessarily of “second category,” meaning it cannot be written as a countable union of nowhere-dense sets. Given a subset A of a complete metric space X, showing that A is of first category is a mathematically precise way of demonstrating that A constitutes a very minor portion of the set X. The term “meager” is often used to mean a set of first category.With the stage set, we now outline the argument that continuous functionsthat are differentiable at even one point of [0,1] form a meager subset of the metric space C[0, 1].
Complete the proof of the theorem. This result is called the Baire Category Theorem because it creates two categories of size for subsets in a metric space. A set of “first category” is one that can be written as a countable union of nowhere-dense sets. These are the small, intuitively thin subsets of a metric space. We now see that if our metric space is complete, then it is necessarily of “second category,” meaning it cannot be written as a countable union of nowhere-dense sets. Given a subset A of a complete metric space X, showing that A is of first category is a mathematically precise way of demonstrating that A constitutes a very minor portion of the set X. The term “meager” is often used to mean a set of first category.
With the stage set, we now outline the argument that continuous functions
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The Baire Category Theorem is a result that classifies subsets of a metric space into two categories based on their size. A subset of "first category" is one that can be written as a countable union of nowhere-dense sets, and is considered to be small or thin. A subset of "second category" is one that cannot be written as a countable union of nowhere-dense sets and is considered to be large or thick. If a metric space is complete, then it is necessarily of second category. The Baire Category Theorem is used to show that certain subsets of a complete metric space, such as the set of continuous functions that are differentiable at even one point of the interval [0,1], are of first category and therefore constitute a very small portion of the space.
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