By Symposium in Pure Mathematics Stanford University 1976, Visit Amazon's R. James Milgram Page, search results, Learn about Author Central, R. James Milgram, , American Mathematical Society

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Una sana regola che costa poco e che aiuta ad evitare i pi` u classici paradossi `e questa: non dire l’insieme degli insiemi tali che . . ma preferire la famiglia degli insiemi . . ; questo eviter`a inoltre monotone ripetizioni. La stessa regola si applica alle famiglie e quindi diremo: la classe delle famiglie. . , la collezione delle classi . . e cos`ı via. 1 Notazioni e riscaldamento Se X `e un insieme scriveremo x ∈ X se x appartiene a X, cio`e se x `e un elemento di X. Indicheremo con ∅ l’insieme vuoto, mentre i simboli {∗} e {∞} denoteranno entrambi la singoletta, ossia l’insieme formato da un solo elemento.

1. Sia U ∈ I(x) un intorno fissato. Allora tutti gli intorni di x contenuti in U formano un sistema fondamentale di intorni di x. 2. Se B `e una base della topologia, allora gli aperti di B che contengono x formano un sistema fondamentale di intorni di x. 12. 6 (♥). Siano A, B sottoinsiemi di uno spazio topologico. Dimostrare che vale A ∪ B = A ∪ B. 7. Sia A un sottoinsieme denso di uno spazio topologico X; dimostrare che per ogni aperto U ⊂ X vale U ⊂ U ∩ A. 8. Sul piano R2 si consideri la famiglia T formata dall’insieme vuoto, da R2 e da tutti i dischi aperti {x2 + y2 < r 2 }, per r > 0.

Si dimostra facilmente che ogni catena ammette un maggiorante e quindi per il lemma di Zorn esiste un elemento massimale (A, f). Dimostriamo che |A| = |X|: se per assurdo |A| < |X|, allora X −A sarebbe infinito e quindi conterrebbe un sottoinsieme B infinito numerabile. Scelta un’applicazione iniettiva g : B × N → B possiamo definire un’applicazione h : (A∪B)×N → A∪B ponendo h(x, n) = f(x, n) se x ∈ A e h(x, n) = g(x, n) se x ∈ B. L’applicazione h `e iniettiva, estende f e quindi contraddice la massimalit`a di (A, f).

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