![]() However its perhaps easier in this case to apply the compact definition directly by exhibiting an open cover for an unbounded set with no finite subcover. In a complete metric space, a closed set is a set which is closed under the limit operation. begingroup You are using the definition of sequentially compact, which is equivalent to compactness for metric spaces. 1 2 In a topological space, a closed set can be defined as a set which contains all its limit points. Considering a convex cone as an example, we easily see that a complete metric of positive curvature given on a plane may be realized by an unbounded convex. 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. Closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. ![]() Suppose that there is \(x \in U_1 \cap S\) and \(y \in U_2 \cap S\). 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.
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