By Kenji Ueno, Koji Shiga, Shigeyuki Morita

ISBN-10: 0821832824

ISBN-13: 9780821832820

This ebook will convey the sweetness and enjoyable of arithmetic to the school room. It deals critical arithmetic in a full of life, reader-friendly type. integrated are routines and plenty of figures illustrating the most techniques.

The first bankruptcy offers the geometry and topology of surfaces. between different issues, the authors speak about the Poincaré-Hopf theorem on serious issues of vector fields on surfaces and the Gauss-Bonnet theorem at the relation among curvature and topology (the Euler characteristic). the second one bankruptcy addresses a number of elements of the idea that of measurement, together with the Peano curve and the Poincaré method. additionally addressed is the constitution of three-d manifolds. particularly, it really is proved that the 3-dimensional sphere is the union of 2 doughnuts.

This is the 1st of 3 volumes originating from a sequence of lectures given via the authors at Kyoto college (Japan).

**Read Online or Download A mathematical gift, 1, interplay between topology, functions, geometry, and algebra PDF**

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**Sample text**

Since (xn ) does not converge, the sequence (yn ) also does not converge. Then the sets {xn ∶ n ∈ N} and {yn ∶ n ∈ N} are near in the metric proximity but far in the ﬁne proximity, a contradiction. (b) If Y is uniformly discrete for some ε > 0, then, for each n ∈ N, there are distinct points xn , yn ∈ Y such that 0 ≤ d(xn , yn ) ≤ n1 . Then the sets {xn ∶ n ∈ N} and {yn ∶ n ∈ N} are near in the metric proximity. However, since Y has no limit points, they are far in the ﬁne proximity, a contradiction.

The set N of natural numbers is near the set E = {n − n1 ∶ n ∈ N} but the image sets f (N) and f (E) are far from each other, since the gap functional on them equals inf {∣n2 − (n − n1 )2 ∣} = 2. However, the following important result is true (its utility will be shown later). 4. Let (X, d), (Y, d′ ) be metric spaces. Assume X has ﬁne proximity δ0 and Y has any compatible proximity λ and let f ∶ X → Y be a function from X to Y . Then f is continuous, if and only if, f is proximally continuous.

If f is an isometry from space X onto Z, then X is said to be isometrically embedded into Y . Metric Space Completion: Completion of a metric space X is accomplished by isometrically embedding X onto a subset of a metric space known to be complete and then take its closure. Use the fact that R is a complete metric space. Then show that the space C ∗ (X) of bounded, realvalued, continuous functions with the sup metric d′ is complete. The ﬁnal step is to embed X isometrically into C ∗ (X). 23. The metric space (C ∗ (X), d′ ) is complete.

### A mathematical gift, 1, interplay between topology, functions, geometry, and algebra by Kenji Ueno, Koji Shiga, Shigeyuki Morita

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