Problem 6. Give an example of a metric space (X, d) and a (necessarily infinite) family of open sets An C X such that NnENAn is not open and not closed.
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Metric space example
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- 16. The set S = { x∈R: x2 - 4<0} with the usual metric is .......................... A. Compact. B. Connected. C. Not connected. D. Sequentially compact.d((x1, x2, x3), (y1, y2, y3) = |x1 - y1| + |x2 - y2| + |x3 - y3|. Conclude that, with this metric, a subset of ℝ3 is sequentially compact if and only if it is closed and bounded.Suppose (S,d) is a metric space. How can we prove that S is open?
- If (X, τ) is a topological space and if A,B⊂XShow that ∂(A∩B)⊂∂A∪∂B.1. Let (M, d) be a compact metric space. Show that closed subsets of M are compact.Let (X, T ) be a topological space, (M, d) be a complete metric space andBC(X, M) := {f ∈ C(X, M); f[X] is bounded }d∞(f, g) := sup d(f(x), g(x)) (f, g ∈ BC(X, M)).Then (BC(X, M), d∞) is a complete metric space.
- This is a discrete structures questionIf A is 3 × 3 with rank A = 2, show that the dimension of the null space of A is 1.Consider the following statements: (i) If A is not compact then there are infinitely many open covers for A that do not have a finite subcover. (ii) If A is compact there must be an open cover for A that has infinitely many finite subcovers. Which statements are true?