![]() We can assume that \(x x\), then for any \(\delta > 0\) the ball \(B(z,\delta) = (z-\delta,z+\delta)\) contains points that are not in \(U_2\), and so \(z \notin U_2\) as \(U_2\) is open. Suppose that there is \(x \in U_1 \cap S\) and \(y \in U_2 \cap S\). In a complete metric space, a closed set is a set which is closed under the limit operation. 1 2 In a topological space, a closed set can be defined as a set which contains all its limit points. ![]() We will show that \(U_1 \cap S\) and \(U_2 \cap S\) contain a common point, so they are not disjoint, and hence \(S\) must be connected. Closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. ![]() A metric space (M, d) is a bounded metric space. C(K) is a metric space, that convergence with respect to this distance is equivalent to uniform convergence and that as a metric space C(K) is. A subset S of a metric space (M, d) is bounded if there exists r > 0 such that d(s, t) < r for all s and t in S. We have shown previously that this is a distance, i.e. The metric d induced by the norm is defined by d(:c, y) '91 - For 2: e. You should recall that a continuous function on a compact metric space is bounded, so the function d(f g) sup x2K jf(x) g(x)j is well-de ned. ![]() \), \(U_1 \cap S\) and \(U_2 \cap S\) are nonempty, and \(S = \bigl( U_1 \cap S \bigr) \cup \bigl( U_2 \cap S \bigr)\). L2 COMPLETE NORMED LINEAR SPACES I.2.1 Definition. ![]()
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