# A mathematical gift, 2, interplay between topology, by Kenji Ueno, Koji Shiga, Shigeyuki Morita, Toshikazu Sunada

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

This booklet brings the wonder and enjoyable of arithmetic to the study room. It deals severe arithmetic in a full of life, reader-friendly kind. incorporated are routines and lots of figures illustrating the most innovations.

The first bankruptcy talks in regards to the conception of trigonometric and elliptic services. It contains topics akin to strength sequence expansions, addition and multiple-angle formulation, and arithmetic-geometric capability. the second one bankruptcy discusses a number of features of the Poncelet Closure Theorem. This dialogue illustrates to the reader the assumption of algebraic geometry as a style of learning geometric homes of figures utilizing algebra as a device.

This is the second one of 3 volumes originating from a sequence of lectures given via the authors at Kyoto collage (Japan). it truly is appropriate for school room use for prime institution arithmetic academics and for undergraduate arithmetic classes within the sciences and liberal arts. the 1st quantity is obtainable as quantity 19 within the AMS sequence, Mathematical international. a 3rd quantity is drawing close.

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Extra info for A mathematical gift, 2, interplay between topology, functions, geometry, and algebra

Example text

We write I = [0, 1] for the unit interval. 6. 1 Cylinder. Here we identify (t, 0) with (t, 1). The quotient space is I × I/∂I ∼ = I × S 1 . In figure (1) A = D and B = C after the identification. 2 M¨ obius band. Here we identify (t, 0) with (1 − t, 1). A realization as a subset of R3 is shown in the figures ?? and ??. After the identification A = C and B = D. M can also be defined as the quotient of S 1 × [−1, 1] with respect to (z, t) ∼ (−z, −t). In this way it is obtained as a quotient of the cylinder S 1 × [−1, 1].

2) Let f ∈ C(X) and ε > 0 be given. We show that there exists g ∈ A such that the inequalities f − ε < g < f + ε hold. By (1) we can choose for each pair x, y ∈ X a function hx,y ∈ A such that hx,y (x) = f (x) and hx,y (y) = f (y). Each y ∈ X has an open neighbourhood Uy such that for z ∈ Uy the inequality hx,y (z) < f (z) + ε holds. Let Uy(1) , . . , Uy(n) be a covering of X. By our additional assumption, the minimum hx of the hx,y(j) is contained in A and satisfies hx < f + ε as well as hx (x) = f (x).

Let A ⊂ R and CA = A × [0, 1]/A × 1 (the formal cone on A). The map f : CA → R2 , (a, t) → ((1 − t)a, t) is injective and continuous. If A is compact, then f is an embedding. If A = ∅ is an open interval, then f is not an embedding (consider neighbourhoods of the cone point A × 1; it has no countable neighbourhood basis). 10. Let (fn : X → R) be a sequence of continuous functions on a compact space X. Suppose that for t ∈ X and n ∈ N the inequality fn (t) ≤ fn+1 (t) holds. If the sequence converges pointwise to a continuous function f , then the convergence is uniform (Theorem of Dini ).