A General Topology Workbook by Iain T. Adamson

By Iain T. Adamson

This paintings goals to offer uncomplicated topology in an unconventional means. It supplies a evaluate of the fundamental definitions including workouts with out strategies or proofs of the theorems partly 1, after which provides the ideas partially 2, permitting the coed to check solutions with their very own.

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26 esiste un sottoinsieme C ⊂ A che `e infinito numerabile. Osserviamo che C ∪ B `e numerabile, ossia ha la stessa cardinalit`a di C e quindi A = C ∪ (A − C) ha la stessa cardinalit` a di A ∪ B = (C ∪ B) ∪ (A − C). 28. Per ogni insieme infinito X vale |X × N| = |X|. Dimostrazione. Consideriamo la famiglia A delle coppie (E, f) tali che E ⊂ X e f : E × N → E iniettiva. Poich´e sappiamo che X contiene sottoinsiemi infiniti numerabili la famiglia A non `e vuota. Ordiniamo A per estensione, cio`e (E, f) ≤ (H, g) se e solo se E ⊂ H e g estende f.

L’insieme dei numeri naturali (interi positivi). N0 = {0, 1, 2, 3, . } l’insieme degli interi non negativi. Z = {0, ±1, ±2, ±3, . } l’anello degli interi. Z/n il gruppo delle classi di resto modulo n. Q, R e C i campi dei numeri razionali, reali e complessi. Assumeremo che il lettore abbia familiarit` a con il principio di induzione e le altre propriet` a dei numeri naturali. Principio di induzione. Sia P : N → {vero, falso} un’applicazione tale che P (1) = vero e tale che P (n) = vero ogni volta che P (n − 1) = vero.

Dimostrare che f `e continua se e solo se M (k) e m(k) sono aperti per ogni k. 17. Due sottoinsiemi A, B di uno spazio topologico si dicono aderenti se (A ∩ B) ∪ (A ∩ B) = ∅. Dimostrare che un’applicazione `e continua se e solo se preserva la relazione di aderenza tra sottoinsiemi. 18. Provare che composizione di omeomorfismi `e un omeomorfismo, che l’inverso di un omeomorfismo `e un omeomorfismo e quindi che l’insieme Omeo(X) degli omeomorfismi di uno spazio topologico X in s´e `e un gruppo. Mostrare inoltre che la relazione sulla categoria2 degli spazi topologici, X ∼ Y se e solo se X `e omeomorfo a Y , `e una relazione di equivalenza (detta equivalenza topologica).

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