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1. G is a topological group and H is a subgroup of G. Show that the projection G -> G/H is open.
Solution


2. Give an example of two compact subspaces of a non-Hausdorff space, whose intersection is not compact.
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3. A complete metric space X is of category II (by Baire's theorem), meaning it cannot be written as a countable union of nowhere dense subsets of itself. Show that every open subset of X (viewed as a topological subspace) is also of category II.
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4. Show that there exists a closed set S in R^2 such that the convex hull conv(S) is not closed.
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5. Let X be a separable metric space and f: X -> R a function such that limit(f(x),a) exists for every a in X. Show that the set of discontinuities of f is at most countable.
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6. Suppose X is an uncountable set and d is any metric on X which induces the discrete topology there. Show that there exist epsilon>0 and an uncountable subset A C X such that d(x,y) >= epsilon for all x,y in A with x <> y.
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7. Show that a map f: X -> Y is continuous if and only if for every A C X, f(Cl(A)) C Cl(f(A)), where Cl(A) denotes the closure of A.
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