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Mathematics W4051x Topology Assignment #6 Due October 21, 2011 1. (a) Let S and T be two topologies on the same set X with S ⊂ T . What does compactness of X under one of these topologies imply about compactness under the other? Give proofs or counterexamples. (b) Show that if X is compact Hausdorff under both S and T , then S = T . 2. (a) Show that any topological space with the cofinite topology is compact. (b) Let the cocountable topology on R be the topology under which U ⊂ R is open if and only if either U = ∅ or its complement is countable. Show that R with the cocountable topology is not compact. 3. A subset C ⊂ X of a metric space X is said to be bounded if there exists x0 ∈ X and d0 ∈ R such that for all x ∈ C, d(x0 , x) ≤ d0 . (a) Show that every compact subspace of a metric space is closed and bounded. (b) Give an example of a metric space in which not every closed and bounded subset is compact. 4. (20 pts) Let {An | n ∈ N} be a countable family of compact, connected subsets of a Hausdorff space X such that An ⊃ An+1 for all n ∈ N. Let A= \ An . n∈N Prove that (a) A is nonempty if and only if each An is nonempty; (b) A is compact; (c) A is connected. Hint for (a): construct an open cover of A1 and pass to a finite subcover. Hint for (c): if A = C ∪ D with C, D clopen, disjoint, and nonempty, choose disjoint T open sets U, V ⊂ X containing C and D respectively, and show that n∈N (An \ (U ∪V )) is nonempty.