Download A Mathematical Gift, 1: The Interplay Between Topology, by Kenji Ueno, Koji Shiga, Shigeyuki Morita, Toshikazu Sunada PDF

By Kenji Ueno, Koji Shiga, Shigeyuki Morita, Toshikazu Sunada

This booklet will deliver the wonder and enjoyable of arithmetic to the study room. It bargains critical arithmetic in a full of life, reader-friendly sort. incorporated are routines and lots of figures illustrating the most strategies.

The first bankruptcy offers the geometry and topology of surfaces. between different subject matters, the authors speak about the Poincaré-Hopf theorem on severe 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 numerous points of the concept that of measurement, together with the Peano curve and the Poincaré process. additionally addressed is the constitution of three-d manifolds. particularly, it really is proved that the third-dimensional sphere is the union of 2 doughnuts.

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

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Extra resources for A Mathematical Gift, 1: The Interplay Between Topology, Functions, Geometry, and Algebra

Example text

Show that Q is countably infinite. 2. Show that the maps f and g of Examples 1 and 2 are bijections. 3. Let X be the two-element set {0, 1}. Show there is a bijective correspondence between the set P (Z+ ) and the cartesian product X ω . 4. (a) A real number x is said to be algebraic (over the rationals) if it satisfies some polynomial equation of positive degree x n + an−1 x n−1 + · · · + a1 x + a0 = 0 with rational coefficients ai . Assuming that each polynomial equation has only finitely many roots, show that the set of algebraic numbers is countable.

There exists a unique element of R called one, different from 0 and denoted by 1, such that x · 1 = x for all x ∈ R. (4) For each x in R, there exists a unique y in R such that x + y = 0. For each x in R different from 0, there exists a unique y in R such that x · y = 1. (5) x · (y + z) = (x · y) + (x · z) for all x, y, z ∈ R. A Mixed Algebraic and Order Property (6) If x > y, then x + z > y + z. If x > y and z > 0, then x · z > y · z. Order Properties (7) The order relation < has the least upper bound property.

That is, A is finite if it is empty or if there is a bijection f : A −→ {1, . . , n} for some positive integer n. In the former case, we say that A has cardinality 0; in the latter case, we say that A has cardinality n. For instance, the set {1, . . , n} itself has cardinality n, for it is in bijective correspondence with itself under the identity function. 37 40 Set Theory and Logic Ch. 1 Now note carefully: We have not yet shown that the cardinality of a finite set is uniquely determined by the set.

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