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MATH 701 - ELEMENTARY TOPOLOGY I Exercise list 4 1. Let S 1 = {(x, y) ∈ R2 x2 +y 2 = 1} be the circle, with the Euclidean metric induced from R2 . Prove that there does not exist any proper subset A $ S 1 which is homeomorphic to S 1 . 2. Let {Vk }k∈N be a collection of non-empty subsets of Rn satisfying, for all k ∈ N, (a) Vk is closed and bounded; and (b) Vk+1 ⊆ Vk . Prove that ∩N Vk is a non-empty compact subset of Rn . 3. Let p : (X, TX ) → (Y, TY ) be a continuous, surjective map that sends closed subsets to closed subsets. Suppose in addition that Y is compact and, for each y ∈ Y the subset p−1 ({y}) is compact in X . Prove that X is compact (hint: if U is an open set containing p−1 ({y}), check that there is an open subset W containing y and such that p−1 (W ) is contained in U ). 4. Let (X, TX ) be a compact, metrizable space. Show that X is second-countable. 5. Let f : (X, TX ) → (Y, TY ) be a continuous map that sends open subsets to open subsets. (a) Show that if X is first-countable (respectively second-countable) then f (X) is firstcountable (respectively second-countable). (b) Give an example of a continuous function f : (X, TX ) → (Y, TY ) such that X is first-countable but f (X) is not first-countable. 6. Let (X, TX ) be a topological space. Prove that ∆ = {(x, x) | x ∈ X} ⊆ X × X is closed (in the product topology) if and only if X is Hausdorff. 7. Let X = R and let b = {[a, b) | a, b ∈ R, a < b}. This turns out to be a basis for a topology on X , T . Let L = {(x, −x) | x ∈ X} ⊆ X × X . Prove the following. (a) L is closed in the product topology. (b) L has the discrete topology as a subspace of X × X . 8. Let f, g : (X, TX ) → (Y, TY ) be continuous functions, and suppose Y is Hausdorff. Show that {x | f (x) = g(x)} ⊆ X is closed. 9. Let p : X → Y be a closed (i.e., p(A) ⊆ Y is closed for all A ⊆ X closed), continuous, surjective map. Suppose in addition that p−1 ({y}) is compact for all y ∈ Y . Show the following. (a) If X is Hausdorff, so is Y . (b) If Y is regular, so is Y . 1 10. A space (X, TX ) is completely normal if every subspace is normal. Show that (X, TX ) is completely normal if and only if the following holds: • For all A, B ⊆ X satisfying ClA ∩ B = ∅ = A ∩ ClB , there exist U, V ∈ TX such that A ⊆ U , B ⊆ V and U ∩ V = ∅. Hint: if X is completely normal, consider (ClA ∩ ClB)c ⊆ X . 11. Let (X, TX ) be a compact Hausdorff space. Show that X is metrizable if and only if X is second countable. 12. Let (X, TX ) be a compact Hausdorff space that is the union of two closed subsets, X1 and X2 . Show that if X1 and X2 are metrizable, then so is X (hint: construct a countable collection A of open sets of X whose intersection with Xi for a basis for Xi , for i = 1, 2; assume X1 \ X2 and X2 \ X1 belong to A; let b be the family of finite intersections of elements of A.) 2