MATH527_HW1 - Serge Ballif MATH 527 Homework 1 September 7...

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Serge Ballif MATH 527 Homework 1 September 7, 2007 Problem 1. Let X be a topological space, and A be its subset. Is it true that A 0 , the set of limit points of A is always closed in X ? It is not true in general. For a counterexample, we can take X = { a, b } with the indiscrete topology T = { X, ∅} . A point x in X is a limit point of a subset A iff every open set containing x also contains a point of A other than x itself. Consider the subset A = { a } . The point b is a limit point of A , because every open set containing b has A as a subset. However, a is not a limit point of A , since no open set containing a contains any point of A other than a . Thus A 0 = { b } , which is not closed in X . Problem 2. Construct a map f : that is continuous at exactly one point. Let f : be given by f ( x ) = ( x if x is rational - x if x is irrational We claim that f is continuous only at the point x = 0. A function f : is continuous at a point x iff for every sequence ( x i ) i =1 that converges to x we also have the sequence ( f ( x i )) i =1 converging to f ( x ). Let
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