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Igor Zelenko, Fall 2009 1 Homework Assignment 1 in Topology I, MATH636 due to Sept 7, 2009 1. a. Find an example of a metric space (X, d) containing two points x1 6= x2 such that for some real r > 0 the ball of radius r with center at x1 coincides with the ball of radius r with center at x2 , i.e. Br (x1 ) = Br (x2 ); b. Prove or disprove by giving a counter-example: If d1 and d2 are metrics on a set X, then the function d(x, y) = min{d1 (x, y), d2 (x, y)} defines a metric on X. c. Show that if (X, d) is a metric space, then ρ(x, y) = d(x,y) 1+d(x,y) is a metric in X and ρ and d are equivalent metric, i.e. they induce the same topology on X. 2. a. Let X be any set, and define Tf = {U ⊂ X : X\U is finite} ∪ {∅}. Show that Tf is a topology on X. b. Is the collection T∞ = {U ⊂ X : X\U is infinite} ∪ {X, ∅} a topology on X? Give a proof to your answer. 3. A map f : X → Y between two topological spaces is open if whenever U ⊂ X is open, then f (U ) ⊂ Y is open in Y . Consider X = R2 with the lexicographic (dictionary) order (cf. p. 6, example 6 in the text) and R with its usual topology. Let f : X → R and g : X → R be the projection onto the first and second factors, respectively. (That is, f (x, y) = x and g(x, y) = y.) Show that f is not an open map and g is an open map. 4. Let D denote the unit ball B1 (0) centered at 0 ∈ R2 with the Euclidean metric. a. Show that D is homeomorphic to R2 . b. Show that D is homeomorphic to the upper half plane H = {(x, y) ∈ R2 : y > 0}.