By Kenji Ueno, Koji Shiga, Shigeyuki Morita

This booklet will deliver the wonder and enjoyable of arithmetic to the study room. It deals critical arithmetic in a full of life, reader-friendly variety. integrated are routines and plenty of figures illustrating the most ideas.

The first bankruptcy provides 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 variety of points of the idea that of size, together with the Peano curve and the Poincaré method. additionally addressed is the constitution of third-dimensional 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 sequence of lectures given via 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**

Suppose C is closed in the union UD n , where each D.. is perfectly normal and closed in X. First let f1 be any nonnegative continuous function on D1 zero exactly on C nD1. I i

Then ((,(B l ) CXn-l(l(X -A), and fJ(B l ) C p(Xn-l(l(X -A)). Also fJ(B 2 ) =p«({'(En)(lA) =f«({,(En)(lA) C Yn-l becausef is cellular. Thus fJ(En) is contained in a union of cells of dimension less than n. If fJ = pif;, then if; (En) C Y n-l because (Y, 'lr) is a cell structure. This completes the first part of the theorem. The statement about strict equivalence is very easy and will be omitted. 2. We denote the cell complex obtained from cell complexes (X,s) and (Y,::I) by adjunction of X to Y as above by (YV, X, SV,::I) and call it the adjunction complex.

2 (1) each T i is closed in X, and the finite union Ui T i is closed in X. Thus crnA = crn (U i Ti) is closed in cr. Since X has the weak topology with respect to s, A is closed in X. Since the subspace A CX is Hausdorff and properties (i) and (iii) are obviously inherited by the sub complex (A,::I), to prove (2) we need only check property (ii) for (A,::I). If A is given the relative topology and F is closed in A, then F is closed in X because, by (1), A is closed. But then F ncr is closed in cr for each cr E S, and, in particular, F ncr is closed in cr for each cr E::I.