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).

**Read Online or Download A Mathematical Gift, 1: The Interplay Between Topology, Functions, Geometry, and Algebra PDF**

**Similar topology books**

**Download PDF by John W. Milnor: Topology from the Differentiable Viewpoint**

LOC 65-26874

This based booklet through uncommon mathematician John Milnor, presents a transparent and succinct advent to at least one of an important matters in glossy arithmetic. starting with uncomplicated techniques corresponding to diffeomorphisms and delicate manifolds, he is going directly to research tangent areas, orientated manifolds, and vector fields. Key thoughts resembling homotopy, the index variety of a map, and the Pontryagin development are mentioned. the writer provides proofs of Sard's theorem and the Hopf theorem.

**New PDF release: Topological Vector Spaces: Chapters 1-5**

It is a softcover reprint of the English translation of 1987 of the second one variation of Bourbaki's Espaces Vectoriels Topologiques (1981).

This Äsecond editionÜ is a new publication and entirely supersedes the unique model of approximately 30 years in the past. yet most of the fabric has been rearranged, rewritten, or changed through a extra up to date exposition, and a great deal of new fabric has been included during this ebook, all reflecting the growth made within the box over the last 3 decades.

Table of Contents.

Chapter I: Topological vector areas over a valued field.

Chapter II: Convex units and in the community convex spaces.

Chapter III: areas of continuing linear mappings.

Chapter IV: Duality in topological vector spaces.

Chapter V: Hilbert areas (elementary theory).

Finally, there are the standard "historical note", bibliography, index of notation, index of terminology, and a listing of a few very important homes of Banach areas.

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).

**Download PDF by N. Bourbaki: General Topology: Chapters 1–4**

This is often the softcover reprint of the English translation of 1971 (available from Springer due to the fact that 1989) of the 1st four chapters of Bourbaki's Topologie générale. It provides all of the fundamentals of the topic, ranging from definitions. vital sessions of topological areas are studied, uniform constructions are brought and utilized to topological teams.

- Explorations in Topology: Map Coloring, Surfaces and Knots
- Writing the Structures of the Subject: Lacan and Topology
- The Cube-A Window to Convex and Discrete Geometry
- Topology: An Introduction
- Algebraic Topology, Homotopy and Homology
- Topological Vector Spaces: Chapters 1–5

**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.