Help me answer number 9. (9) Show that the given collection F is an open cover for S such that it does not contain a finite subcover and soS is not compact.(a) S = (0, 2); and F = { U₂ | n € N } where Un = (1, 2-1⁄2)(b) S = (0, ∞); and F = { Un | n ≤ N} where Un = (0, n)

a cover of a set {\displaystyle X} is a collection of sets whose union includes {\displaystyle X} as…

Answered: Show that the given collection F is an…

Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Let (X,T) be a topological space, K a compact subset of X and (Fn)n∈ N a family in P(X) with Fn≠ø for all n. If K⊇F̅0⊇F̅1⊇....⊇F̅n⊇........ then prove that (by way of contradiction) n=0∩∞ F̅n ≠ø
3. a) We know that the set S = {1/n : n ∈ N} is not compact because 0 is a limit point of S that is not in S. To see the non-compactness of S in another way, find an open cover of S that does not have a finite subcover! b) Is the Cantor set closed? Is it compact? Explain!
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Show that the given collection F is an open cover for S such that it does not contain a finite subcover and so S is not compact. (a) S = (0, 2); and F = { U₂ | n ¤ N } where Un = (1, 2-1) (b) S = (0, ∞); and F = { Un | n € N} where Un = (0, n)