By Kenji Ueno, Koji Shiga, Shigeyuki Morita

ISBN-10: 0821832824

ISBN-13: 9780821832820

This ebook will carry the sweetness and enjoyable of arithmetic to the study room. It deals critical arithmetic in a full of life, reader-friendly kind. integrated are routines and plenty of figures illustrating the most strategies.

The first bankruptcy offers the geometry and topology of surfaces. between different themes, 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 variety of facets of the idea that of size, together with the Peano curve and the Poincaré technique. additionally addressed is the constitution of third-dimensional manifolds. specifically, it's proved that the third-dimensional sphere is the union of 2 doughnuts.

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

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**Additional info for A mathematical gift, 1, interplay between topology, functions, geometry, and algebra**

**Example text**

F [univ M(X )] ⊂ univ M(Y ). f [univ N(X )] ⊂ univ N(Y ). f [univ N(X )] ⊂ univ M(Y ). Proof. 43 yields the equivalence of the first three conditions since f is a Borel measurable map. 50. 58 gives condition (4) implies condition (5). Obviously, condition (5) implies condition (6). It remains to prove that condition (6) implies condition (2). Suppose that condition (2) fails. 49, condition (6) fails. This completes the proof. 3. Universally positive closure. Let M be a subset of a separable metrizable space X .

Proof. 4. Grzegorek’s cardinal number κG 23 (x∗ , y∗ ) in {0, 1}N × {0, 1}N and such that the diagram X Θ f −−−−→ Y ϑ f∗ {0, 1}N × {0, 1}N −−−−→ {0, 1}N commutes, that is, f B f ∗ (see page 195 for the definition of the B-successor relation N N and Y ∗ = ϑ {0, 1}N . 48 there ∗ B ). Let X = Θ {0, 1} × {0, 1} is a continuous bijection g : N → Y0 of an absolute null space N contained in {0, 1}N onto a non-absolute measurable space Y0 contained in {0, 1}N . 25, we have that graph(g) is an absolute null space.

The use of the continuum hypothesis in the early years of the subject of absolute null spaces will be commented on below. Absolute null spaces have played a special role in what is called the Ulam numbers. 2 it was shown that ℵ1 is an Ulam number. Other remarks on Ulam numbers will be given below in the Comment section. 3. 5. Existence under the continuum hypothesis. There is a novel presentation of the existence of absolute null spaces with the aid of the continuum hypothesis in the book Measure and Category by J.

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

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