By Aiyalam P Balachandran, Giuseppe Marmo, Bo-Sture Skagerstam

Balachandran, Marmo, Skagerstam, Stern. Classical topology and quantum states (WS, 1991)(ISBN 9810203292)(KA)(T)(375s)_PQgf_

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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 ).