By Kenji Ueno, Koji Shiga, Shigeyuki Morita

This ebook will convey the sweetness and enjoyable of arithmetic to the school room. It deals critical arithmetic in a full of life, reader-friendly type. integrated are routines and plenty of figures illustrating the most techniques.

The first bankruptcy offers the geometry and topology of surfaces. between different subject matters, the authors talk 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. particularly, it's proved that the 3-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 collage (Japan).

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**Extra info for A Mathematical Gift III: The Interplay Between Topology, Functions, Geometry, and Algebra (Mathematical World, Volume 23)**

**Example text**

3) Every mapping of a discrete space into a topological space is continuous. 1 Let f be a mapping of a topological space X into a topological space X'. Then the following statements are equivalent: a) f is continuous in X. b) For every subset A of X, feA) cf(A). c) The inverse image under f of every closed subset of X' is a closed subset of X. d) The inverse image under I of every open subset of X' is an open subset of X. THEOREM I. We have already seen that a) implies b) (Proposition I). To show that b) implies c), let F' be a closed subset of X' and let F = (F'); then --1 by hypothesis f(F) cl(F) c F' = F', hence F c I (F') = F c F, so that F = F and F is closed.

Consisting of the elements x for which there is an element y e X", which belongs to the equivalence class of X. Clearly to each x e corresponds a unique y e AI'). " ~ A",). " (i) For each i. , \'0, v) of L and each x e AA" n AAY, we have h",,(x) e AfLY and Converse~, suppose that for each pair of indices (I" Vo) we are given a subset AAIL of Xl. and a mapping hloLA: AAI' ~ AfLA satisfying the conditions (i) and (ii) above. It follows first of all from (ii) applied to the triples (I" \'0, ),) and (Vo.

Chapter VI, § 1). Likewise, starting from the rational line Q, we define the rational number space oj n dimensions Qn (rational plane for n = 2). The topology of the space Rn has as a base the set of all products of n open intervals in R, which are called open boxes oj n dimensions. The open boxes which contain a point xeR" form a fundamental system of neighbourhoods of this point. Likewise the products of n closed intervals in R are called closed boxes of n dimensions. The closed boxes to which x is interior also form a fundamental system of neighbourhoods of x.