MATH 202 A - Problem Set 11
نویسنده
چکیده
proof E is a Borel set: for each n ∈ N, On is open as the union of open sets. Therefore E is the countable intersection of open sets, thus is a Borel set. E is of the second category: we first prove that O n is nowhere dense. Indeed, we have for all qk ∈ Q, B(qk, 1 n2k ) ⊂ On, thus qk ∈ int(On). Therefore Q ⊆ int(On), therefore R = cl(Q) ⊆ cl(int(On)), i.e. cl(int(On)) = R. Taking complements, and using the fact that for any set X, int(X) = cl(X) and cl(X) = int(X), we have int(cl(O n)) = ∅ i.e. O n is nowhere dense. Now we have R = E ∪ E = E ∪ (∪n=1O n) therefore E is not of the first category, otherwise we would have R is of the first category (countable union of nowhere dense sets), which contradicts Baire’s theorem. m(E) = 0. We prove that for all > 0, m(E) ≤ . Let > 0, and let n0 ∈ N such that 4/n0 ≤ . Then we have
منابع مشابه
The Extent of Using Real Problems and Emphasizing Modelling in the 10th Grade Math Textbook
Modelling is emphasized heavily in the national math curriculum, yet the extent to which this concept is covered in the math textbooks is, while of great importance, unknown. To discover the extent to which this issue, and real problems in general, are covered in the 10th grade math textbooks, the content of the textbook for the ‘theoretical’ (“nazari”) branch was analyzed. In doing so, the vie...
متن کاملFUZZY BOUNDED SETS AND TOTALLY FUZZY BOUNDED SETS IN I-TOPOLOGICAL VECTOR SPACES
In this paper, a new definition of fuzzy bounded sets and totallyfuzzy bounded sets is introduced and properties of such sets are studied. Thena relation between totally fuzzy bounded sets and N-compactness is discussed.Finally, a geometric characterization for fuzzy totally bounded sets in I- topologicalvector spaces is derived.
متن کاملMATH 202 B - Problem Set 10
(10.1) Let k ∈ N, and let X = C k ([0, 1] d) to be the set of all functions f ∈ C 0 ([0, 1] d) such that for all α ∈ ∆ k , f (α) = ∂f ∂x α 1 1 ...∂x α d d exists and is continuous on (0, 1) d , and extends continuously on [0, 1] d. Here
متن کاملThe Effects of Learning Self-Regulated Learning Strategies on High School Students\' Math Problem-Solving Skills
The Effects of Learning Self-Regulated Learning Strategies on High School Students' Math Problem-Solving Skills B. Shaalchi, Ph.D. Y. Rezaapoor J. Baabaapoor Kheyrodin, Ph.D. M. Alizaad Sardroodi F. Abdollaahi To see the effects of learning self-regulated learning strategies on math problem-solving skills among high school students, a sample of 271 randomly selected h...
متن کاملAn Efficient Genetic Algorithm to Solve the Intermodal Terminal Location problem
The exponential growth of the flow of goods and passengers, fragility of certain products and the need for the optimization of transport costs impose on carriers to use more and more multimodal transport. In addition, the need for intermodal transport policy has been strongly driven by environmental concerns and to benefit from the combination of different modes of transport to cope with the in...
متن کامل