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

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

This ebook will carry the wonder and enjoyable of arithmetic to the school room. It bargains severe arithmetic in a full of life, reader-friendly variety. integrated are routines and lots of figures illustrating the most techniques.

The first bankruptcy offers the geometry and topology of surfaces. between different themes, 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 a number of facets of the idea that of size, together with the Peano curve and the Poincaré technique. additionally addressed is the constitution of 3-dimensional manifolds. specifically, it truly is proved that the three-d 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 collage (Japan).

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Download e-book for kindle: A Mathematical Gift, 1: The Interplay Between Topology, by Kenji Ueno, Koji Shiga, Shigeyuki Morita, Toshikazu Sunada

This ebook will deliver the sweetness and enjoyable of arithmetic to the study room. It deals severe arithmetic in a full of life, reader-friendly kind. incorporated are workouts and lots of figures illustrating the most strategies. the 1st bankruptcy offers the geometry and topology of surfaces. between different themes, the authors talk 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).

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

Example text

Also the universal property of M follows analogously to the case of the product topology. Thus it remains to check uniqueness: let M be another topology on M with the universal property as in (ii). Then consider the identity map of M which is continuous by the universal property of M . But we can exchange M by M in (i) showing that also id : (M, M ) −→ (M, M) is continuous. Thus M = M follows. 3 is called the initial topology with respect to the maps pi : M −→ Mi . The argument concerning the uniqueness of the initial topology can be used at many other places where certain objects are characterized by universal properties in categories.

Since A◦ is the largest open set inside A, we can apply this to A◦ and get that (A◦ )◦ is the largest open set inside A◦ . Since A◦ is already open, we have (A◦ )◦ = A◦ . Analogously, we can argue for the closure to get (Acl )cl = Acl . 9, (iii), the relation ∂(∂ A) = (∂ A)cl \ (∂ A)◦ = ∂ A \ (∂ A)◦ ⊆ ∂ A, completing the second part. 9, (i) and (ii). For the fourth part we have A◦ ⊆ A ⊆ A ∪ B and also B ◦ ⊆ A ∪ B. e. in (A ∪ B)◦ . Next, A ⊆ A ∪ B implies Acl ⊆ (A ∪ B)cl and analogously B cl ⊆ (A ∪ B)cl showing Acl ∪ B cl ⊆ (A ∪ B)cl .

First we show uniqueness. Thus let M and M be two topologies on M with the above universal property. Then id : (M, M ) −→ (M, M ) is continuous and hence qi : (Mi , Mi ) −→ (M, M ) is continuous. But then also id : (M, M) −→ (M, M ) is continuous by the universal property of M. This means M ⊆ M and by symmetry we have M = M. For the existence define M to be the topology by declaring O ⊆ M to be open if qi−1 (O) is open in Mi for all i ∈ I . Since taking preimages is compatible with arbitrary unions and intersections it follows that M is indeed a topology.

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