Math 190: Fall 2014 Homework 5 Due 5:00pm on Friday 11/14/2014 Problem 1: Let p : X → Y be a continuous map of topological spaces. Suppose that there exists a continuous map f : Y → X such that p ◦ f is the identity map on Y . Prove that p is a quotient map. Problem 2: Let Dn denote the closed unit ball of radius 1 Dn = {(x1 , x2 , . . . , xn ) ∈ Rn : x21 + x22 + · · · + x2n ≤ 1} in Rn centered at the origin. Consider the n − 1-dimensional unit sphere S n−1 ⊂ Dn . Prove that the quotient space Dn /S n−1 obtained by collapsing S n−1 to a point is homeomorphic to S n . Problem 3: (Exercise 22.3 in Munkres) Let π : R2 → R be projection onto the first coordinate. Let A be the subspace of R2 given by those points (x, y) ∈ R2 for which x ≥ 0 or y = 0 (or both). Let p : A → R be the restriction of π1 to A. Prove that p is a quotient map, but is neither open nor closed. (Recall: A map f : X → Y is open if U open implies f (U ) open and closed if C closed implies f (C) closed.) Problem 4: (Exercise 22.5 in Munkres) Let p : X → Y be an open map and let A by a subset of X. Prove that the map q : A → p(A) obtained by restricting p is also an open map. Problem 5: Let ∼ be the equivalence relation on R2 given by (x, y) ∼ (x0 , y 0 ) if y + x2 = y 0 + x02 . The quotient space R2 / ∼ is homeomorphic to a familiar topological space X. Find X and prove your assertion. Problem 6: Let X and Y be topological spaces and let X q Y be the disjoint union of X and Y . Prove that the set {U q V : U open in X and V open in Y } is a topology on X q Y . This is the disjoint union topology. Prove that if Z is any topological space and f : X → Z and g : Y ( → Z are continuous functions, then h : X q Y → Z is also f (a) a ∈ X continuous, where h(a) = . (Category fans: Disjoint union is the “dual g(a) a ∈ Y construction” to product.) Problem 7: Let X1 and X2 be two copies of the real line R. Define an equivalence relation on X1 q X2 by x1 ∼ x2 for all x 6= 0, where x1 is the copy of x in X1 and x2 is the copy of x in X2 . Let X = (X1 q X2 )/ ∼ be the “line with two origins”. Prove that X is not Hausdorff. Problem 8: (Exercise 23.4 in Munkres) Prove that an infinite set X is connected in the finite complement topology. Problem 9: (Exercise 23.5 in Munkres) A space X is called totally disconnected if its only connected subspaces are one-point sets. Show that if X is discrete, then X is totally disconnected. Does the converse hold? 1

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