General topology by Jacques Dixmier

By Jacques Dixmier

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Next, there exists n ~ 2N such that Xn E B(x, 1/2N). Then B(Xn'~) c: B(Xn,2~) C B(X, 2~ + 2~) cUi' which is absurd. (b) It now suffices to prove that X can be covered by a finite number of balls B(x, a). Let Xl E X. If B(XI' a) = X, the proof is over. Otherwise, let X2 EX - B(x l , a). If B(x l , a) v B(X2' a) = X, the proof is over. Otherwise, let X3 E X - (B(XI' a) v B(X2' a»; etc. If the process stops, the theorem is established. Otherwise, there exists a sequence (Xl> X2,"') of points of X such that for every n.

The following conditions are equivalent: E F. (i) W is a neighborhood ofx in F; Oi) W is the intersection with F of a neighborhood of x in E. (i) => (ii). Suppose W is a neighborhood of x in F. There exists an open subset B of F such that x E B c W. Then there exists an open subset A of E such that B = F n A. Let V = A u W. Then x E A c: V, thus V is a neighborhood of x in E. On the other hand, F n V = (F n A) u (F n W) = B u W = W. 1. Topological Subspaces (ii) => (i). Suppose W = F n V, where V is a neighborhood of x in E.

9(i». We have thus defined a topology on X'. The subset X of X' is open in X'. The intersections with X of the open sets of X' are the open sets of X. In other words, the topology induced on X by that of X' is the given topology on X. Let us show that X' is separated. Let x, y be distinct points of X', and let us show that x and y admit disjoint neighborhoods in X'. This is clear if x, y E X. Suppose x = wand y E X. Let W be a compact neighborhood of y in X. This is also a neighborhood of y in X' (because X is open in X').

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