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1. Basic Point Set Topology Consider Rn with its usual topology and its usual vector space structure over R . Let Lin(Rp , Rq ) stand for the set of linear maps from Rp to Rq and observe that it is isomorphic to Rpq with respect to its natural structure as vector space over R . PROBLEMS I. Let X, Y be compact, connected, Hausdorff spaces. (A) Prove that a local homeomorphism f : X −→ Y is a covering projection. (B) Prove that an immersion h : Sn −→ Sn , where n ≥ 2 is a diffeomorphism. What happens if n = 1? II. Let A, X, Y be topological spaces. (A) If X is compact and Y is Hausdorff, prove that every continuous map f : X −→ Y is closed. (B) If a surjective continuous map p : A −→ X is open or closed, prove that p is a quotient map. Give counterexamples to demonstrate that there exist quotient maps that are neither open nor closed. (C) Let q : A −→ X be a quotient map. Prove that g : X −→ Y is continuous iff g ◦ q : A −→ Y is continuous. (D) Let r : A −→ X be an open, surjective map. Prove that h : X −→ Y is open iff h ◦ r : A −→ Y is open. III. Let X be a topological space. Given a group G, a (left) action of G on X (by homeomorphisms) is a continuous map (G assumed to have the discrete topology) ρ : G × X −→ X such that (1) ρ(e, x) = x, (2) ρ(gh, x) = ρ(g, ρ(h, x)) for all g, h ∈ G and x ∈ X. Clearly ρ(g, •) : X −→ X is a homeomorphism and ρ(g, •)−1 = ρ(g −1 , •) for each g ∈ G . In those cases where there is no ambiguity as regards the choice of the action ρ, the notation ρ(g, x) is usually abbreviated into gx. In the presence of a group action on X one introduces the equivalence relation ∼ with respect to which x ∼ y if y = gx for some g ∈ G. By abuse of notation, the set of equivalence classes under ∼ is denoted by X/G . (A) If X/G has the quotient topology induced by the map q : X −→ X/G sending each x ∈ X into the equivalence class containing x, prove that q is an open map. (B) Write Tn , Sn , RP n respectively in the form Rn /G1 , (Rn+1 − {(0, 0, 0)})/G2 , (Rn+1 − {(0, 0, 0)})/G3 by introducing suitable group actions. IV. Let X be a topological space. Let X be a topological space, R ⊆ X × X be an equivalence relation on X. Consider the quotient space X/R and the quotient map q : X −→ X/R. (A) If X/R is Hausdorff, prove that R is a closed subset of X × X. (B) If q is an open map and R is a closed subset of X × X, prove that X/R is Hausdorff. (C) Apply the above results to prove that Tn and RP n are Hausdorff spaces. V. Remember that the (n + 1)-dimensional closed disc En+1 and the n-dimensional sphere Sn are defined by En+1 = {u ∈ Rn+1 | k u k≤ 1} Sn = {u ∈ Rn+1 | k u k= 1} . 1 (A) Let M, N be topological spaces and A, B ⊆ M be closed subsets with M = A ∪ B. Given continuous functions f : A −→ N, g : B −→ N with f |A∩B = g|A∩B , prove that the function h : M −→ N defined by f (m) for m∈A h(m) = g(m) for m∈B is continuous. (B) Let x : dom(x) ⊆op M −→ Rn be a chart. Given a closed set C ⊆ x(dom(x)), clearly x−1 (C) is closed in dom(x). Is x−1 (C) is necessarily closed in M ? Suppose that K ⊆ x(dom(x)) is compact. If M is Hausdorff, prove that x−1 (K) is closed in M. (C) If M is an n-dimensional manifold, prove that there exists a surjective continuous map h : M −→ Sn . (D) Remember that a topological space X is said to be homogeneous if for any a, b ∈ X there exists a homeomorphism h : X −→ X such that h(a) = b. Prove that for any a, b ∈ Int En+1 , there exists a homeomorphism h : En+1 −→ En+1 with h|Sn = IdSn , such that h(a) = b. (E) Let M be a manifold. Prove that each m ∈ M has a neighbourhood U with the property that for each a, b ∈ U there exists a homeomorphism h : M −→ M such that h(a) = b. (F) Prove that a connected manifold is a homogeneous topological space. (G) Give an example of a non-homogeneous manifold. (H) Does there exist a connected, locally Euclidean, homogeneous topological space which is not Hausdorff ? 2