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Week 5 Exercises

Version 2024/10/30. These exercises in PDF To other course material

  • Exercise 5.1. Let \(A\) be a subspace of a topological space \(X.\) Prove: if \(F\subseteq A\) and \(F\) is closed in \(X,\) then \(F\) is closed in \(A.\)

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  • Exercise 5.2 (unions and intersections of compact sets). Let \(X\) be a topological space.

    • 1. Show that a union of two compact subsets of \(X\) is compact.

    • 2. Assuming that \(X\) is Hausdorff, show that an intersection of two compact subsets of \(X\) is compact. (Why do we need \(X\) to be Hausdorff?)

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  • Exercise 5.3 (nested sequence of closed subsets in a compact). Let \(K\) be a compact topological space. Assume that \(F_1\supseteq F_2\supseteq F_3\supseteq \dots ,\) where \(F_i\) is a non-empty closed subset of \(K\) for each \(i\ge 1.\) Prove that all the sets \(F_i\) have a common point.

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  • Exercise 5.4 (a Hausdorff compact topology is “optimal”). Let \((X,\mathscr T)\) be a Hausdorff compact topological space. Use the Topological Inverse Function Theorem to show that

    • 1. any topology on \(X,\) which is strictly weaker than \(\mathscr T,\) is not Hausdorff;

    • 2. any topology on \(X,\) which is strictly stronger than \(\mathscr T,\) is not compact.

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Version 2024/10/30. These exercises in PDF To other course material