Homework 1 Topology I, Fall 2013 Problem 1 (An ultrametric space). Let X be the set of all right-infinite binary words, i.e., X = {x1 x2 x3 · · · | xi ∈ {0, 1}, for i = 1, 2, . . . }. (a) Define dist : X × X → R as follows. For x, y ∈ X, if x 6= y, then dist(x, y) = e−|x∧y| , where x ∧ y denotes the longest common prefix of x and y, and |x ∧ y| denotes its length, and if x = y, then d(x, y) = 0. Show that d is a metric on X. (b) An ultrametric on a set Y is a function d : Y × Y → R that satisfies the following laws: 1. (positivity) d(x, y) ≥ 0 with equality if and only if x = y, 2. (symmetry) d(x, y) = d(y, x), and 3. (ultrametric inequality) d(x, z) ≤ max{d(x, y), d(y, z)}. Note that every ultrametric is a metric (do not show this). Show that if two balls in a metric space Y with an ultrametric d intersect nontrivially then one of them is contained in the other. (c) Show that dist defined in (a) is an ultrametric on X. Problem 2 (Topology forgets distances). Let (X, d) be a metric space. Define d : X × X → R and d0 : X × X → R by d(x, y) = min{1, d(x, y)} and d0 (x, y) = d(x, y) 1 + d(x, y) (a) Show that both d and d0 are metrics on X (note that both of these metrics are bounded; the metric d is called the standard bounded metric induced by d). (b) Show that the topologies on X induced by d, d and d0 are the same (thus, topology “does not remember” if a metric space was bounded or not; also, balls of large radius are not important). Problem 3 (Continuity and closure). Let X and Y be topological spaces and f : X → Y a function. Prove the following statements. (a) f is continuous if and only if, for all A ⊆ X, we have f (A) ⊆ f (A). (b) f is closed if and only if, for all A ⊆ X, we have f (A) ⊇ f (A). Problem 4 (Limit points have large company (well, sometimes)). Let X be a topological space, A ⊆ X, and x a limit point of A (every neighborhood of x contains a point from A − {x}). (a) Show that if X is T1 , then every neighborhood of x contains infinitely many points from A. (b) Provide an example of a T0 space X and a neighborhood of x that contains only finitely many points from A. Problem 5 (Equality is a closed condition (well, sometimes)). Let X and Y be topological spaces and f, g : X → Y two maps between them. (a) Show that, if Y is a Hausdorff space, then the set Xf =g = {x ∈ X | f (x) = g(x)} is closed in X. (b) Provide an example in which Y is not a Hausdorff space and Xf =g is not closed. Problem 6 (Gluing Lemma). Let X and Y be topological spaces, X = ∪j∈J Aj , and {fj : Aj → Y | j ∈ J} a family of maps such that, for all i, j ∈ J for which Ai ∩ Aj 6= ∅, the restrictions fi |Ai ∩Aj and fj |Ai ∩Aj coincide. Define a function f : X → Y by gluing, i.e., whenever x ∈ Aj let f (x) = fj (x). Show that f is continuous under either of the following two conditions. (a) Each Aj is open. (b) Each Aj is closed and J is finite. Problem 7 (If all you ask is T1 , that’s what you get). Let X = (R, τ ), where τ is the smallest topology in which every singleton {x}, for x ∈ R, is closed. Show that X is a T1 space, but not a Hausdorff space. 2