Week 5 Exercises
Version 2024/10/30. These exercises in PDF To other course material
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Exercise 5.2 (unions and intersections of compact sets). Let \(X\) be a topological space.
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1. Show that a union of two compact subsets of \(X\) is compact.
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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 5.4 (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
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1. any topology on \(X,\) which is strictly weaker than \(\mathscr T,\) is not Hausdorff;
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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