By Y. Eliashberg
This e-book offers the 1st steps of a thought of confoliations designed to hyperlink geometry and topology of 3-dimensional touch buildings with the geometry and topology of codimension-one foliations on third-dimensional manifolds. constructing nearly independently, those theories at the start look belonged to 2 diversified worlds: the speculation of foliations is a part of topology and dynamical platforms, whereas touch geometry is the odd-dimensional 'brother' of symplectic geometry. even though, either theories have built a couple of remarkable similarities. Confoliations - which interpolate among touch buildings and codimension-one foliations - may also help us to appreciate greater hyperlinks among the 2 theories. those hyperlinks supply instruments for transporting effects from one box to the other.It's gains comprise: a unified method of the topology of codimension-one foliations and speak to geometry; perception at the geometric nature of integrability; and, new effects, particularly at the perturbation of confoliations into touch buildings
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This booklet provides the 1st steps of a idea of confoliations designed to hyperlink geometry and topology of 3-dimensional touch buildings with the geometry and topology of codimension-one foliations on 3-dimensional manifolds. constructing virtually independently, those theories at the beginning look belonged to 2 varied worlds: the speculation of foliations is a part of topology and dynamical platforms, whereas touch geometry is the odd-dimensional 'brother' of symplectic geometry.
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Sample text
F (n)) per ogni n ≥ 1. ) sono note a tutti. Vediamo adesso un’altra applicazione che sarà utilizzata in seguito. 3. Sia X ⊂ N un sottoinsieme infinito. Allora esiste un’applicazione bigettiva e strettamente crescente f : N → X. Dimostrazione. Per ogni sottoinsieme finito Y ⊂ X il complementare X − Y non è vuoto e quindi ammette minimo. Basta definire f in modo ricorsivo come f (1) = min(X), f (n + 1) = min(X − {f (1), f (2), . . , f (n)}). Per quanto riguarda i numeri reali, risulta essere di particolare importanza il seguente principio.
12 (di Cantor–Schröder–Bernstein). Siano X, Y due insiemi. Se esistono due applicazioni iniettive f : X → Y e g : Y → X, allora X e Y hanno la stessa cardinalità. Dimostrazione. 11 esiste un sottoinsieme A ⊂ X tale che, ponendo B = Y − f (A) vale A ∩ g(B) = ∅ e A ∪ g(B) = X. Per come abbiamo definito B si ha inoltre che B∩f (A) = ∅ e B∪f (A) = Y , mentre dall’iniettività di f e g segue che le due applicazioni f : A → f (A) e g : B → g(B) sono bigettive. Basta adesso osservare che l’applicazione h : X → Y, h(x) = f (x) g −1 se x ∈ A (x) se x ∈ g(B) è bigettiva.
Si osservi che un sottoinsieme B di uno spazio topologico è aperto se e solo se B = B ◦ ed è chiuso se e solo se B = B. Passando al complementare si ottiene la relazione X − B ◦ = X − B. 17. Nella topologia euclidea sulla retta reale R, per ogni a < b si ha: ]a, b[ = [a, b[ = ]a, b] = [a, b], [a, b]◦ = [a, b[◦ =]a, b]◦ = ]a, b[ . 18. La frontiera di un sottoinsieme B di uno spazio topologico è il chiuso ∂B = B − B ◦ = B ∩ X − B. Dunque i punti della frontiera ∂B sono i punti aderenti sia a B che al complementare X − B.